{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\n", "theoremStatement": "lemma 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "PFR.Mathlib.Probability.Kernel.Composition", "Mathlib.Analysis.Calculus.TangentCone", "Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics", "Mathlib.Analysis.Calculus.FDeriv.Basic", 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"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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 66}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\n", "theoremStatement": "lemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B ", "theoremName": "IsShift.card_congr", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 36, "tokenPositionInFile": 1008, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 1, "numPremises": 25, "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", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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", 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"Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by rintro \u27e8x, rfl\u27e9; simp", "proofType": "tactic", "proofLengthLines": 0, "proofLengthTokens": 27}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n", "theoremStatement": "/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' ", "theoremName": "wlog_notInCoset", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]", "proofType": "tactic", "proofLengthLines": 24, "proofLengthTokens": 1358}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n", "theoremStatement": "/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] ", "theoremName": "torsion_free_doubling", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "1226201", "date": "2024-01-02"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 81, "tokenPositionInFile": 3354, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 19, "numPremises": 426, "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", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]", "proofType": "tactic", "proofLengthLines": 117, "proofLengthTokens": 8125}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n", "theoremStatement": "/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] ", "theoremName": "torsion_dist_shrinking", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 203, "tokenPositionInFile": 11772, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 9, "numPremises": 125, "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", 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"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", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 492}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n", "theoremStatement": "/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) ", "theoremName": "app_ent_PFR'", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9", "proofType": "tactic", "proofLengthLines": 36, "proofLengthTokens": 1584}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n", "theoremStatement": "lemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) ", "theoremName": "app_ent_PFR", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "9692d13", "date": "2024-01-11"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 286, "tokenPositionInFile": 15840, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 60, "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", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 96}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n", "theoremStatement": "/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] ", "theoremName": "PFR_projection'", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "9692d13", "date": "2024-01-11"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 294, "tokenPositionInFile": 16290, "theoremPositionInFile": 9}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 435, "importedModules": 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"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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9", "proofType": "tactic", "proofLengthLines": 93, "proofLengthTokens": 4973}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n", "theoremStatement": "/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] ", "theoremName": "PFR_projection", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "9692d13", "date": "2024-01-11"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 398, "tokenPositionInFile": 22003, "theoremPositionInFile": 10}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 8, "numPremises": 281, "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", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 343}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\n", "theoremStatement": "lemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d ", "theoremName": "four_logs", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 420, "tokenPositionInFile": 23051, "theoremPositionInFile": 11}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "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", 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"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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 62}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\n", "theoremStatement": "lemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 ", "theoremName": "sum_prob_preimage", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 424, "tokenPositionInFile": 23253, "theoremPositionInFile": 12}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 107, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 530}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n", "theoremStatement": "/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) ", "theoremName": "single_fibres", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 438, "tokenPositionInFile": 24019, "theoremPositionInFile": 13}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 25, "numPremises": 476, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]", "proofType": "tactic", "proofLengthLines": 131, "proofLengthTokens": 7867}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\n", "theoremStatement": "lemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S ", "theoremName": "exists_coset_cover", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 595, "tokenPositionInFile": 33499, "theoremPositionInFile": 14}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "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", "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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9", "proofType": "tactic", "proofLengthLines": 2, "proofLengthTokens": 103}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\n", "theoremStatement": "lemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S ", "theoremName": "dimension_le_of_coset_cover", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 604, "tokenPositionInFile": 33913, "theoremPositionInFile": 16}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 2, "numPremises": 33, "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", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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", 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"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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 59}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\n", "theoremStatement": "lemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 121}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n", "theoremStatement": "/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n  (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) ", "theoremName": "Finsupp.mapRange_surjective", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 622, "tokenPositionInFile": 34560, "theoremPositionInFile": 18}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "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", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  classical\n  let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n  have : Function.RightInverse g f := by\n    intro n\n    by_cases h : n = 0\n    . simp [g, h, hf]\n    \u00b7 simp [g, h, Function.surjInv_eq hs n]\n  have hg : g 0 = 0 := by simp [g]\n  have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n  intro F\n  use Finsupp.mapRange g hg F\n  rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n  convert Finsupp.mapRange_id F\n  convert Function.RightInverse.id this", "proofType": "tactic", "proofLengthLines": 14, "proofLengthTokens": 461}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n  (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n  classical\n  let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n  have : Function.RightInverse g f := by\n    intro n\n    by_cases h : n = 0\n    . simp [g, h, hf]\n    \u00b7 simp [g, h, Function.surjInv_eq hs n]\n  have hg : g 0 = 0 := by simp [g]\n  have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n  intro F\n  use Finsupp.mapRange g hg F\n  rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n  convert Finsupp.mapRange_id F\n  convert Function.RightInverse.id this\n\n", "theoremStatement": "/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G ", "theoremName": "torsion_free", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "f0dd05d", "date": "2023-12-27"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 640, "tokenPositionInFile": 35316, "theoremPositionInFile": 19}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 223, "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", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n    rintro x hx hn\n    rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n    rcases hn with \u27e8 n, hn, hn' \u27e9\n    apply_fun Module.Free.repr \u2124 G at hn'\n    simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n    linarith", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 262}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n  (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n  classical\n  let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n  have : Function.RightInverse g f := by\n    intro n\n    by_cases h : n = 0\n    . simp [g, h, hf]\n    \u00b7 simp [g, h, Function.surjInv_eq hs n]\n  have hg : g 0 = 0 := by simp [g]\n  have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n  intro F\n  use Finsupp.mapRange g hg F\n  rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n  convert Finsupp.mapRange_id F\n  convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n    rintro x hx hn\n    rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n    rcases hn with \u27e8 n, hn, hn' \u27e9\n    apply_fun Module.Free.repr \u2124 G at hn'\n    simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n    linarith\n\n", "theoremStatement": "/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n.  Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n  let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n  ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) ", "theoremName": "weak_PFR_quotient_prelim", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "f0dd05d", "date": "2023-12-27"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 649, "tokenPositionInFile": 35684, "theoremPositionInFile": 20}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 280, "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", 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"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", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  have hH_elem : ElementaryAddCommGroup H 2 := by\n    apply ElementaryAddCommGroup.quotient_group (by decide)\n    intro x; rw [AddMonoidHom.mem_range]\n    use x\n    rw [zsmulAddGroupHom_apply]\n    norm_cast\n  let B := Module.Free.ChooseBasisIndex \u2124 G\n  let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n  have hB_fin : Fintype B := by infer_instance\n  have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n    rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n  have hH_module : Module (ZMod 2) H := by infer_instance\n  let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n  let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n  have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n    intro x hx\n    simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n    simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n      Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n    rcases hx with \u27e8y, rfl\u27e9\n    ext b\n    simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n      Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n    left\n    exact ZMod.nat_cast_self 2\n  let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n  have hsur : Function.Surjective g := by\n    have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n    have h2 := h1.comp bG.repr.surjective\n    have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n      ext x b\n      simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n        Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n      rw [QuotientAddGroup.lift_mk]\n      simp [mod, f]\n    rw [h3] at h2\n    apply Function.Surjective.of_comp h2\n  have hinj : Function.Injective g := by\n    rw [injective_iff_map_eq_zero]\n    intro x hx\n    rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n    simp only [QuotientAddGroup.mk'_apply, g] at hx\n    rw [QuotientAddGroup.lift_mk] at hx\n    simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n      Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n      ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n    replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n    let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n    let z' := (Finsupp.equivFunOnFinite).symm z\n    change \u2200 b : B, 2 * z' b = (f y) b at hx\n    let x' := bG.repr.symm z'\n    rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n    simp_rw [\u03c8, zsmulAddGroupHom_apply]\n    use x'\n    change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n    rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n    rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n    change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n    congr\n    ext b\n    rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n  rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n  have bH : Basis B (ZMod 2) H := by\n    constructor\n    exact {\n      toFun := g\n      invFun := g'\n      left_inv := hg'.1\n      right_inv := hg'.2\n      map_add' := AddMonoidHom.map_add _\n      map_smul' := by\n        intro r x\n        rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n        change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n        rw [intCast_smul, intCast_smul]\n        exact AddMonoidHom.map_zsmul g x n\n    }\n  have hH_fin : Fintype H := Module.fintypeOfFintype bH\n  have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n    rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n    congr\n  exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9", "proofType": "tactic", "proofLengthLines": 87, "proofLengthTokens": 4047}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n  (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n  classical\n  let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n  have : Function.RightInverse g f := by\n    intro n\n    by_cases h : n = 0\n    . simp [g, h, hf]\n    \u00b7 simp [g, h, Function.surjInv_eq hs n]\n  have hg : g 0 = 0 := by simp [g]\n  have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n  intro F\n  use Finsupp.mapRange g hg F\n  rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n  convert Finsupp.mapRange_id F\n  convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n    rintro x hx hn\n    rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n    rcases hn with \u27e8 n, hn, hn' \u27e9\n    apply_fun Module.Free.repr \u2124 G at hn'\n    simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n    linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n.  Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n  let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n  ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  have hH_elem : ElementaryAddCommGroup H 2 := by\n    apply ElementaryAddCommGroup.quotient_group (by decide)\n    intro x; rw [AddMonoidHom.mem_range]\n    use x\n    rw [zsmulAddGroupHom_apply]\n    norm_cast\n  let B := Module.Free.ChooseBasisIndex \u2124 G\n  let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n  have hB_fin : Fintype B := by infer_instance\n  have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n    rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n  have hH_module : Module (ZMod 2) H := by infer_instance\n  let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n  let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n  have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n    intro x hx\n    simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n    simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n      Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n    rcases hx with \u27e8y, rfl\u27e9\n    ext b\n    simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n      Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n    left\n    exact ZMod.nat_cast_self 2\n  let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n  have hsur : Function.Surjective g := by\n    have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n    have h2 := h1.comp bG.repr.surjective\n    have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n      ext x b\n      simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n        Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n      rw [QuotientAddGroup.lift_mk]\n      simp [mod, f]\n    rw [h3] at h2\n    apply Function.Surjective.of_comp h2\n  have hinj : Function.Injective g := by\n    rw [injective_iff_map_eq_zero]\n    intro x hx\n    rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n    simp only [QuotientAddGroup.mk'_apply, g] at hx\n    rw [QuotientAddGroup.lift_mk] at hx\n    simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n      Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n      ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n    replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n    let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n    let z' := (Finsupp.equivFunOnFinite).symm z\n    change \u2200 b : B, 2 * z' b = (f y) b at hx\n    let x' := bG.repr.symm z'\n    rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n    simp_rw [\u03c8, zsmulAddGroupHom_apply]\n    use x'\n    change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n    rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n    rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n    change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n    congr\n    ext b\n    rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n  rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n  have bH : Basis B (ZMod 2) H := by\n    constructor\n    exact {\n      toFun := g\n      invFun := g'\n      left_inv := hg'.1\n      right_inv := hg'.2\n      map_add' := AddMonoidHom.map_add _\n      map_smul' := by\n        intro r x\n        rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n        change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n        rw [intCast_smul, intCast_smul]\n        exact AddMonoidHom.map_zsmul g x n\n    }\n  have hH_fin : Fintype H := Module.fintypeOfFintype bH\n  have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n    rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n    congr\n  exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n", "theoremStatement": "/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n  let H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x ", "theoremName": "third_iso", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 743, "tokenPositionInFile": 40069, "theoremPositionInFile": 21}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 97, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  have h1 : G\u2082 \u2264 N := by\n    intro x hx\n    rw [\u2190 eq_zero_iff] at hx\n    have : \u03c6 x = 0 := hx\n    simp [N, this, AddSubgroup.zero_mem H']\n  set H'' := AddSubgroup.map (mk' G\u2082) N\n  have h2 : H' = H'' := by\n    change H' = AddSubgroup.map (mk' G\u2082) N\n    rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n    simp\n  let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n  let e2 := quotientAddEquivOfEq h2\n  set e := e2.trans e1\n  use e\n  intro x\n  convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1", "proofType": "tactic", "proofLengthLines": 19, "proofLengthTokens": 648}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n  (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n  classical\n  let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n  have : Function.RightInverse g f := by\n    intro n\n    by_cases h : n = 0\n    . simp [g, h, hf]\n    \u00b7 simp [g, h, Function.surjInv_eq hs n]\n  have hg : g 0 = 0 := by simp [g]\n  have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n  intro F\n  use Finsupp.mapRange g hg F\n  rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n  convert Finsupp.mapRange_id F\n  convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n    rintro x hx hn\n    rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n    rcases hn with \u27e8 n, hn, hn' \u27e9\n    apply_fun Module.Free.repr \u2124 G at hn'\n    simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n    linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n.  Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n  let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n  ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  have hH_elem : ElementaryAddCommGroup H 2 := by\n    apply ElementaryAddCommGroup.quotient_group (by decide)\n    intro x; rw [AddMonoidHom.mem_range]\n    use x\n    rw [zsmulAddGroupHom_apply]\n    norm_cast\n  let B := Module.Free.ChooseBasisIndex \u2124 G\n  let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n  have hB_fin : Fintype B := by infer_instance\n  have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n    rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n  have hH_module : Module (ZMod 2) H := by infer_instance\n  let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n  let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n  have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n    intro x hx\n    simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n    simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n      Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n    rcases hx with \u27e8y, rfl\u27e9\n    ext b\n    simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n      Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n    left\n    exact ZMod.nat_cast_self 2\n  let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n  have hsur : Function.Surjective g := by\n    have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n    have h2 := h1.comp bG.repr.surjective\n    have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n      ext x b\n      simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n        Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n      rw [QuotientAddGroup.lift_mk]\n      simp [mod, f]\n    rw [h3] at h2\n    apply Function.Surjective.of_comp h2\n  have hinj : Function.Injective g := by\n    rw [injective_iff_map_eq_zero]\n    intro x hx\n    rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n    simp only [QuotientAddGroup.mk'_apply, g] at hx\n    rw [QuotientAddGroup.lift_mk] at hx\n    simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n      Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n      ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n    replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n    let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n    let z' := (Finsupp.equivFunOnFinite).symm z\n    change \u2200 b : B, 2 * z' b = (f y) b at hx\n    let x' := bG.repr.symm z'\n    rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n    simp_rw [\u03c8, zsmulAddGroupHom_apply]\n    use x'\n    change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n    rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n    rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n    change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n    congr\n    ext b\n    rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n  rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n  have bH : Basis B (ZMod 2) H := by\n    constructor\n    exact {\n      toFun := g\n      invFun := g'\n      left_inv := hg'.1\n      right_inv := hg'.2\n      map_add' := AddMonoidHom.map_add _\n      map_smul' := by\n        intro r x\n        rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n        change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n        rw [intCast_smul, intCast_smul]\n        exact AddMonoidHom.map_zsmul g x n\n    }\n  have hH_fin : Fintype H := Module.fintypeOfFintype bH\n  have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n    rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n    congr\n  exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n  let H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  have h1 : G\u2082 \u2264 N := by\n    intro x hx\n    rw [\u2190 eq_zero_iff] at hx\n    have : \u03c6 x = 0 := hx\n    simp [N, this, AddSubgroup.zero_mem H']\n  set H'' := AddSubgroup.map (mk' G\u2082) N\n  have h2 : H' = H'' := by\n    change H' = AddSubgroup.map (mk' G\u2082) N\n    rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n    simp\n  let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n  let e2 := quotientAddEquivOfEq h2\n  set e := e2.trans e1\n  use e\n  intro x\n  convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\n", "theoremStatement": "lemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n    [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n    z \u2208 A ", "theoremName": "single", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 769, "tokenPositionInFile": 41187, "theoremPositionInFile": 22}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 98, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  contrapose! hz\n  have : Disjoint {z} A := by simp [hz]\n  replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n  simp [hA] at this\n  have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n  simp [this] at h\n  assumption", "proofType": "tactic", "proofLengthLines": 7, "proofLengthTokens": 266}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n  (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n  classical\n  let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n  have : Function.RightInverse g f := by\n    intro n\n    by_cases h : n = 0\n    . simp [g, h, hf]\n    \u00b7 simp [g, h, Function.surjInv_eq hs n]\n  have hg : g 0 = 0 := by simp [g]\n  have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n  intro F\n  use Finsupp.mapRange g hg F\n  rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n  convert Finsupp.mapRange_id F\n  convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n    rintro x hx hn\n    rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n    rcases hn with \u27e8 n, hn, hn' \u27e9\n    apply_fun Module.Free.repr \u2124 G at hn'\n    simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n    linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n.  Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n  let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n  ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  have hH_elem : ElementaryAddCommGroup H 2 := by\n    apply ElementaryAddCommGroup.quotient_group (by decide)\n    intro x; rw [AddMonoidHom.mem_range]\n    use x\n    rw [zsmulAddGroupHom_apply]\n    norm_cast\n  let B := Module.Free.ChooseBasisIndex \u2124 G\n  let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n  have hB_fin : Fintype B := by infer_instance\n  have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n    rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n  have hH_module : Module (ZMod 2) H := by infer_instance\n  let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n  let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n  have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n    intro x hx\n    simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n    simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n      Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n    rcases hx with \u27e8y, rfl\u27e9\n    ext b\n    simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n      Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n    left\n    exact ZMod.nat_cast_self 2\n  let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n  have hsur : Function.Surjective g := by\n    have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n    have h2 := h1.comp bG.repr.surjective\n    have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n      ext x b\n      simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n        Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n      rw [QuotientAddGroup.lift_mk]\n      simp [mod, f]\n    rw [h3] at h2\n    apply Function.Surjective.of_comp h2\n  have hinj : Function.Injective g := by\n    rw [injective_iff_map_eq_zero]\n    intro x hx\n    rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n    simp only [QuotientAddGroup.mk'_apply, g] at hx\n    rw [QuotientAddGroup.lift_mk] at hx\n    simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n      Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n      ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n    replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n    let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n    let z' := (Finsupp.equivFunOnFinite).symm z\n    change \u2200 b : B, 2 * z' b = (f y) b at hx\n    let x' := bG.repr.symm z'\n    rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n    simp_rw [\u03c8, zsmulAddGroupHom_apply]\n    use x'\n    change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n    rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n    rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n    change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n    congr\n    ext b\n    rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n  rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n  have bH : Basis B (ZMod 2) H := by\n    constructor\n    exact {\n      toFun := g\n      invFun := g'\n      left_inv := hg'.1\n      right_inv := hg'.2\n      map_add' := AddMonoidHom.map_add _\n      map_smul' := by\n        intro r x\n        rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n        change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n        rw [intCast_smul, intCast_smul]\n        exact AddMonoidHom.map_zsmul g x n\n    }\n  have hH_fin : Fintype H := Module.fintypeOfFintype bH\n  have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n    rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n    congr\n  exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n  let H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  have h1 : G\u2082 \u2264 N := by\n    intro x hx\n    rw [\u2190 eq_zero_iff] at hx\n    have : \u03c6 x = 0 := hx\n    simp [N, this, AddSubgroup.zero_mem H']\n  set H'' := AddSubgroup.map (mk' G\u2082) N\n  have h2 : H' = H'' := by\n    change H' = AddSubgroup.map (mk' G\u2082) N\n    rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n    simp\n  let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n  let e2 := quotientAddEquivOfEq h2\n  set e := e2.trans e1\n  use e\n  intro x\n  convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n    [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n    z \u2208 A := by\n  contrapose! hz\n  have : Disjoint {z} A := by simp [hz]\n  replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n  simp [hA] at this\n  have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n  simp [this] at h\n  assumption\n\n", "theoremStatement": "/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n    \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n    Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n    By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n    (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n      40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n      \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) ", "theoremName": "weak_PFR_asymm_prelim", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", 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"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", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have : Nonempty A := hnA.to_subtype\n  have : Nonempty B := hnB.to_subtype\n  obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  let _mH : MeasurableSpace H := \u22a4\n  have h_fintype : Fintype H := Fintype.ofFinite H\n  have h_torsionfree := torsion_free (G := G)\n\n  obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n  obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n    exists_isUniform_measureSpace' B\n\n  rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n  let N := AddSubgroup.comap \u03c6 H'\n  set \u03c6' := QuotientAddGroup.mk' N\n  have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n  have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n    constructor\n    intro x\n    exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n  rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n  rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n    \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n  have : Nonempty Ax := hnAx.to_subtype\n  have : Nonempty By := hnBy.to_subtype\n  have Axf : Finite Ax := by rw [hAx]; infer_instance\n  have Byf : Finite By := by rw [hBy]; infer_instance\n\n  have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n  have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n  rw [rdist_symm] at h2\n  rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n  -- using explicit .toFun casts as this saves a lot of heartbeats\n  change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n  change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n  replace hH1 : log (Nat.card H') \u2264 40  * d\u1d64[A # B] := by\n    apply hH1.trans\n    linarith\n  replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n    rw [mul_comm, \u2190 log_rpow (by norm_num)]\n    norm_cast\n    classical\n    rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n    all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n  use N, x, y, Ax, By\n  refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n  replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n    set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n    set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n    have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n    have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n    change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n    have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n    have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n    have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n    rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n    have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] :=  rdist_of_inj hX hY e.toAddMonoidHom he_inj\n    rwa [this]\n\n  set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n  set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n  have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n  have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n  rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n  swap\n  . rw [\u2190 h] at hH2\n    have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n    have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n    have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n    have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n    have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n      change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n      set Af := A.toFinite.toFinset\n      have hUAf : IsUniform Af UA := by\n        convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n      have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UA \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n      replace this := single ((\u2119).map UA) hx this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hxx : Ax = A := by\n      have h : hnAx.some \u2208 Ax := hnAx.some_mem\n      simp [hAx] at h \u22a2\n      have := hAAx h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hAAx hz\n\n    have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n      change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n      set Bf := B.toFinite.toFinset\n      have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n      have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UB \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n      replace this := single ((\u2119).map UB) hy this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hyy : By = B := by\n      have h : hnBy.some \u2208 By := hnBy.some_mem\n      simp [hBy] at h \u22a2\n      have := hBBy h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hBBy hz\n\n    simp [hxx, hyy]\n\n  have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n      congr\n      convert (four_logs ?_ ?_ ?_ ?_).symm\n      all_goals norm_cast; exact Nat.card_pos\n    _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n    _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n      apply mul_le_mul_of_nonneg_right hH2\n      have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n      replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n      rw [\u2190 mul_le_mul_left this]\n      apply le_trans _ hcard_ineq\n      rw [mul_zero]\n      change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n      rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n      apply Real.log_nonneg\n      rw [one_le_div]\n      gcongr\n      . apply Nat.card_mono\n        . exact Set.toFinite A\n        rw [hAx]; exact Set.inter_subset_left _ _\n      apply Nat.card_mono\n      . exact Set.toFinite B\n      rw [hBy]; exact Set.inter_subset_left _ _\n      norm_cast\n      exact mul_pos Nat.card_pos Nat.card_pos\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n      rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n  exact (mul_le_mul_left h).mp this", "proofType": "tactic", "proofLengthLines": 164, "proofLengthTokens": 7954}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n  (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n  classical\n  let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n  have : Function.RightInverse g f := by\n    intro n\n    by_cases h : n = 0\n    . simp [g, h, hf]\n    \u00b7 simp [g, h, Function.surjInv_eq hs n]\n  have hg : g 0 = 0 := by simp [g]\n  have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n  intro F\n  use Finsupp.mapRange g hg F\n  rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n  convert Finsupp.mapRange_id F\n  convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n    rintro x hx hn\n    rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n    rcases hn with \u27e8 n, hn, hn' \u27e9\n    apply_fun Module.Free.repr \u2124 G at hn'\n    simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n    linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n.  Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n  let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n  ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  have hH_elem : ElementaryAddCommGroup H 2 := by\n    apply ElementaryAddCommGroup.quotient_group (by decide)\n    intro x; rw [AddMonoidHom.mem_range]\n    use x\n    rw [zsmulAddGroupHom_apply]\n    norm_cast\n  let B := Module.Free.ChooseBasisIndex \u2124 G\n  let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n  have hB_fin : Fintype B := by infer_instance\n  have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n    rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n  have hH_module : Module (ZMod 2) H := by infer_instance\n  let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n  let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n  have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n    intro x hx\n    simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n    simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n      Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n    rcases hx with \u27e8y, rfl\u27e9\n    ext b\n    simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n      Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n    left\n    exact ZMod.nat_cast_self 2\n  let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n  have hsur : Function.Surjective g := by\n    have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n    have h2 := h1.comp bG.repr.surjective\n    have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n      ext x b\n      simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n        Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n      rw [QuotientAddGroup.lift_mk]\n      simp [mod, f]\n    rw [h3] at h2\n    apply Function.Surjective.of_comp h2\n  have hinj : Function.Injective g := by\n    rw [injective_iff_map_eq_zero]\n    intro x hx\n    rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n    simp only [QuotientAddGroup.mk'_apply, g] at hx\n    rw [QuotientAddGroup.lift_mk] at hx\n    simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n      Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n      ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n    replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n    let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n    let z' := (Finsupp.equivFunOnFinite).symm z\n    change \u2200 b : B, 2 * z' b = (f y) b at hx\n    let x' := bG.repr.symm z'\n    rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n    simp_rw [\u03c8, zsmulAddGroupHom_apply]\n    use x'\n    change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n    rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n    rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n    change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n    congr\n    ext b\n    rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n  rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n  have bH : Basis B (ZMod 2) H := by\n    constructor\n    exact {\n      toFun := g\n      invFun := g'\n      left_inv := hg'.1\n      right_inv := hg'.2\n      map_add' := AddMonoidHom.map_add _\n      map_smul' := by\n        intro r x\n        rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n        change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n        rw [intCast_smul, intCast_smul]\n        exact AddMonoidHom.map_zsmul g x n\n    }\n  have hH_fin : Fintype H := Module.fintypeOfFintype bH\n  have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n    rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n    congr\n  exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n  let H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  have h1 : G\u2082 \u2264 N := by\n    intro x hx\n    rw [\u2190 eq_zero_iff] at hx\n    have : \u03c6 x = 0 := hx\n    simp [N, this, AddSubgroup.zero_mem H']\n  set H'' := AddSubgroup.map (mk' G\u2082) N\n  have h2 : H' = H'' := by\n    change H' = AddSubgroup.map (mk' G\u2082) N\n    rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n    simp\n  let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n  let e2 := quotientAddEquivOfEq h2\n  set e := e2.trans e1\n  use e\n  intro x\n  convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n    [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n    z \u2208 A := by\n  contrapose! hz\n  have : Disjoint {z} A := by simp [hz]\n  replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n  simp [hA] at this\n  have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n  simp [this] at h\n  assumption\n\n/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n    \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n    Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n    By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n    (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n      40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n      \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) := by\n  have : Nonempty A := hnA.to_subtype\n  have : Nonempty B := hnB.to_subtype\n  obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  let _mH : MeasurableSpace H := \u22a4\n  have h_fintype : Fintype H := Fintype.ofFinite H\n  have h_torsionfree := torsion_free (G := G)\n\n  obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n  obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n    exists_isUniform_measureSpace' B\n\n  rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n  let N := AddSubgroup.comap \u03c6 H'\n  set \u03c6' := QuotientAddGroup.mk' N\n  have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n  have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n    constructor\n    intro x\n    exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n  rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n  rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n    \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n  have : Nonempty Ax := hnAx.to_subtype\n  have : Nonempty By := hnBy.to_subtype\n  have Axf : Finite Ax := by rw [hAx]; infer_instance\n  have Byf : Finite By := by rw [hBy]; infer_instance\n\n  have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n  have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n  rw [rdist_symm] at h2\n  rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n  -- using explicit .toFun casts as this saves a lot of heartbeats\n  change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n  change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n  replace hH1 : log (Nat.card H') \u2264 40  * d\u1d64[A # B] := by\n    apply hH1.trans\n    linarith\n  replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n    rw [mul_comm, \u2190 log_rpow (by norm_num)]\n    norm_cast\n    classical\n    rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n    all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n  use N, x, y, Ax, By\n  refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n  replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n    set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n    set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n    have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n    have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n    change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n    have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n    have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n    have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n    rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n    have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] :=  rdist_of_inj hX hY e.toAddMonoidHom he_inj\n    rwa [this]\n\n  set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n  set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n  have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n  have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n  rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n  swap\n  . rw [\u2190 h] at hH2\n    have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n    have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n    have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n    have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n    have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n      change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n      set Af := A.toFinite.toFinset\n      have hUAf : IsUniform Af UA := by\n        convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n      have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UA \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n      replace this := single ((\u2119).map UA) hx this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hxx : Ax = A := by\n      have h : hnAx.some \u2208 Ax := hnAx.some_mem\n      simp [hAx] at h \u22a2\n      have := hAAx h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hAAx hz\n\n    have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n      change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n      set Bf := B.toFinite.toFinset\n      have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n      have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UB \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n      replace this := single ((\u2119).map UB) hy this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hyy : By = B := by\n      have h : hnBy.some \u2208 By := hnBy.some_mem\n      simp [hBy] at h \u22a2\n      have := hBBy h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hBBy hz\n\n    simp [hxx, hyy]\n\n  have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n      congr\n      convert (four_logs ?_ ?_ ?_ ?_).symm\n      all_goals norm_cast; exact Nat.card_pos\n    _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n    _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n      apply mul_le_mul_of_nonneg_right hH2\n      have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n      replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n      rw [\u2190 mul_le_mul_left this]\n      apply le_trans _ hcard_ineq\n      rw [mul_zero]\n      change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n      rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n      apply Real.log_nonneg\n      rw [one_le_div]\n      gcongr\n      . apply Nat.card_mono\n        . exact Set.toFinite A\n        rw [hAx]; exact Set.inter_subset_left _ _\n      apply Nat.card_mono\n      . exact Set.toFinite B\n      rw [hBy]; exact Set.inter_subset_left _ _\n      norm_cast\n      exact mul_pos Nat.card_pos Nat.card_pos\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n      rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n  exact (mul_le_mul_left h).mp this\n\n/-- Separating out the conclusion of `weak_PFR_asymm` for convenience of induction arguments.-/\ndef WeakPFRAsymmConclusion (A B : Set G) : Prop :=\n  \u2203 A' B' : Set G, A' \u2286 A \u2227 B' \u2286 B \u2227 A'.Nonempty \u2227 B'.Nonempty \u2227\n  log (((Nat.card A) * (Nat.card B)) / ((Nat.card A') * (Nat.card B'))) \u2264 34 * d\u1d64[A # B] \u2227\n  max (dimension A') (dimension B') \u2264 (40 / log 2) * d\u1d64[A # B]\n\n/-- The property of two sets A,B of a group G not being contained in cosets of the same proper subgroup -/\ndef not_in_coset {G: Type u} [AddCommGroup G] (A B : Set G) : Prop := AddSubgroup.closure ((A-A) \u222a (B-B)) = \u22a4\n\n\n", "theoremStatement": "/-- In fact one has equality here, but this is tricker to prove and not needed for the argument. -/\nlemma dimension_of_shift {G: Type u} [AddCommGroup G]\n  {H: AddSubgroup G} (A : Set H) (x : G) :\n  dimension ((fun a:H \u21a6 (a:G) + x) '' A) \u2264 dimension A ", "theoremName": "dimension_of_shift", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c8331ef", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  classical\n  rcases Nat.find_spec (exists_coset_cover A) with \u27e8 S, v, hrank, hshift \u27e9\n  change FiniteDimensional.finrank \u2124 S = dimension A at hrank\n  rw [\u2190 hrank]\n  convert dimension_le_of_coset_cover _ (Submodule.map H.subtype.toIntLinearMap S) (x+v) ?_\n  . apply LinearEquiv.finrank_eq\n    exact Submodule.equivMapOfInjective _ (by simpa using Subtype.val_injective) _\n  intro a ha\n  rw [Set.mem_image] at ha\n  rcases ha with \u27e8 b, \u27e8 hb, hb'\u27e9 \u27e9\n  rw [Submodule.mem_map]\n  use b - v, hshift b hb\n  simp [\u2190 hb']\n  abel", "proofType": "tactic", "proofLengthLines": 14, "proofLengthTokens": 524}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n  (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n  classical\n  let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n  have : Function.RightInverse g f := by\n    intro n\n    by_cases h : n = 0\n    . simp [g, h, hf]\n    \u00b7 simp [g, h, Function.surjInv_eq hs n]\n  have hg : g 0 = 0 := by simp [g]\n  have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n  intro F\n  use Finsupp.mapRange g hg F\n  rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n  convert Finsupp.mapRange_id F\n  convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n    rintro x hx hn\n    rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n    rcases hn with \u27e8 n, hn, hn' \u27e9\n    apply_fun Module.Free.repr \u2124 G at hn'\n    simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n    linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n.  Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n  let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n  ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  have hH_elem : ElementaryAddCommGroup H 2 := by\n    apply ElementaryAddCommGroup.quotient_group (by decide)\n    intro x; rw [AddMonoidHom.mem_range]\n    use x\n    rw [zsmulAddGroupHom_apply]\n    norm_cast\n  let B := Module.Free.ChooseBasisIndex \u2124 G\n  let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n  have hB_fin : Fintype B := by infer_instance\n  have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n    rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n  have hH_module : Module (ZMod 2) H := by infer_instance\n  let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n  let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n  have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n    intro x hx\n    simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n    simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n      Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n    rcases hx with \u27e8y, rfl\u27e9\n    ext b\n    simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n      Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n    left\n    exact ZMod.nat_cast_self 2\n  let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n  have hsur : Function.Surjective g := by\n    have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n    have h2 := h1.comp bG.repr.surjective\n    have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n      ext x b\n      simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n        Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n      rw [QuotientAddGroup.lift_mk]\n      simp [mod, f]\n    rw [h3] at h2\n    apply Function.Surjective.of_comp h2\n  have hinj : Function.Injective g := by\n    rw [injective_iff_map_eq_zero]\n    intro x hx\n    rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n    simp only [QuotientAddGroup.mk'_apply, g] at hx\n    rw [QuotientAddGroup.lift_mk] at hx\n    simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n      Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n      ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n    replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n    let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n    let z' := (Finsupp.equivFunOnFinite).symm z\n    change \u2200 b : B, 2 * z' b = (f y) b at hx\n    let x' := bG.repr.symm z'\n    rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n    simp_rw [\u03c8, zsmulAddGroupHom_apply]\n    use x'\n    change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n    rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n    rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n    change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n    congr\n    ext b\n    rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n  rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n  have bH : Basis B (ZMod 2) H := by\n    constructor\n    exact {\n      toFun := g\n      invFun := g'\n      left_inv := hg'.1\n      right_inv := hg'.2\n      map_add' := AddMonoidHom.map_add _\n      map_smul' := by\n        intro r x\n        rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n        change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n        rw [intCast_smul, intCast_smul]\n        exact AddMonoidHom.map_zsmul g x n\n    }\n  have hH_fin : Fintype H := Module.fintypeOfFintype bH\n  have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n    rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n    congr\n  exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n  let H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  have h1 : G\u2082 \u2264 N := by\n    intro x hx\n    rw [\u2190 eq_zero_iff] at hx\n    have : \u03c6 x = 0 := hx\n    simp [N, this, AddSubgroup.zero_mem H']\n  set H'' := AddSubgroup.map (mk' G\u2082) N\n  have h2 : H' = H'' := by\n    change H' = AddSubgroup.map (mk' G\u2082) N\n    rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n    simp\n  let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n  let e2 := quotientAddEquivOfEq h2\n  set e := e2.trans e1\n  use e\n  intro x\n  convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n    [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n    z \u2208 A := by\n  contrapose! hz\n  have : Disjoint {z} A := by simp [hz]\n  replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n  simp [hA] at this\n  have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n  simp [this] at h\n  assumption\n\n/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n    \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n    Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n    By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n    (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n      40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n      \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) := by\n  have : Nonempty A := hnA.to_subtype\n  have : Nonempty B := hnB.to_subtype\n  obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  let _mH : MeasurableSpace H := \u22a4\n  have h_fintype : Fintype H := Fintype.ofFinite H\n  have h_torsionfree := torsion_free (G := G)\n\n  obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n  obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n    exists_isUniform_measureSpace' B\n\n  rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n  let N := AddSubgroup.comap \u03c6 H'\n  set \u03c6' := QuotientAddGroup.mk' N\n  have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n  have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n    constructor\n    intro x\n    exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n  rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n  rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n    \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n  have : Nonempty Ax := hnAx.to_subtype\n  have : Nonempty By := hnBy.to_subtype\n  have Axf : Finite Ax := by rw [hAx]; infer_instance\n  have Byf : Finite By := by rw [hBy]; infer_instance\n\n  have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n  have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n  rw [rdist_symm] at h2\n  rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n  -- using explicit .toFun casts as this saves a lot of heartbeats\n  change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n  change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n  replace hH1 : log (Nat.card H') \u2264 40  * d\u1d64[A # B] := by\n    apply hH1.trans\n    linarith\n  replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n    rw [mul_comm, \u2190 log_rpow (by norm_num)]\n    norm_cast\n    classical\n    rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n    all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n  use N, x, y, Ax, By\n  refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n  replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n    set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n    set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n    have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n    have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n    change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n    have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n    have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n    have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n    rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n    have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] :=  rdist_of_inj hX hY e.toAddMonoidHom he_inj\n    rwa [this]\n\n  set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n  set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n  have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n  have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n  rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n  swap\n  . rw [\u2190 h] at hH2\n    have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n    have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n    have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n    have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n    have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n      change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n      set Af := A.toFinite.toFinset\n      have hUAf : IsUniform Af UA := by\n        convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n      have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UA \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n      replace this := single ((\u2119).map UA) hx this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hxx : Ax = A := by\n      have h : hnAx.some \u2208 Ax := hnAx.some_mem\n      simp [hAx] at h \u22a2\n      have := hAAx h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hAAx hz\n\n    have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n      change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n      set Bf := B.toFinite.toFinset\n      have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n      have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UB \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n      replace this := single ((\u2119).map UB) hy this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hyy : By = B := by\n      have h : hnBy.some \u2208 By := hnBy.some_mem\n      simp [hBy] at h \u22a2\n      have := hBBy h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hBBy hz\n\n    simp [hxx, hyy]\n\n  have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n      congr\n      convert (four_logs ?_ ?_ ?_ ?_).symm\n      all_goals norm_cast; exact Nat.card_pos\n    _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n    _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n      apply mul_le_mul_of_nonneg_right hH2\n      have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n      replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n      rw [\u2190 mul_le_mul_left this]\n      apply le_trans _ hcard_ineq\n      rw [mul_zero]\n      change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n      rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n      apply Real.log_nonneg\n      rw [one_le_div]\n      gcongr\n      . apply Nat.card_mono\n        . exact Set.toFinite A\n        rw [hAx]; exact Set.inter_subset_left _ _\n      apply Nat.card_mono\n      . exact Set.toFinite B\n      rw [hBy]; exact Set.inter_subset_left _ _\n      norm_cast\n      exact mul_pos Nat.card_pos Nat.card_pos\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n      rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n  exact (mul_le_mul_left h).mp this\n\n/-- Separating out the conclusion of `weak_PFR_asymm` for convenience of induction arguments.-/\ndef WeakPFRAsymmConclusion (A B : Set G) : Prop :=\n  \u2203 A' B' : Set G, A' \u2286 A \u2227 B' \u2286 B \u2227 A'.Nonempty \u2227 B'.Nonempty \u2227\n  log (((Nat.card A) * (Nat.card B)) / ((Nat.card A') * (Nat.card B'))) \u2264 34 * d\u1d64[A # B] \u2227\n  max (dimension A') (dimension B') \u2264 (40 / log 2) * d\u1d64[A # B]\n\n/-- The property of two sets A,B of a group G not being contained in cosets of the same proper subgroup -/\ndef not_in_coset {G: Type u} [AddCommGroup G] (A B : Set G) : Prop := AddSubgroup.closure ((A-A) \u222a (B-B)) = \u22a4\n\n\n/-- In fact one has equality here, but this is tricker to prove and not needed for the argument. -/\nlemma dimension_of_shift {G: Type u} [AddCommGroup G]\n  {H: AddSubgroup G} (A : Set H) (x : G) :\n  dimension ((fun a:H \u21a6 (a:G) + x) '' A) \u2264 dimension A := by\n  classical\n  rcases Nat.find_spec (exists_coset_cover A) with \u27e8 S, v, hrank, hshift \u27e9\n  change FiniteDimensional.finrank \u2124 S = dimension A at hrank\n  rw [\u2190 hrank]\n  convert dimension_le_of_coset_cover _ (Submodule.map H.subtype.toIntLinearMap S) (x+v) ?_\n  . apply LinearEquiv.finrank_eq\n    exact Submodule.equivMapOfInjective _ (by simpa using Subtype.val_injective) _\n  intro a ha\n  rw [Set.mem_image] at ha\n  rcases ha with \u27e8 b, \u27e8 hb, hb'\u27e9 \u27e9\n  rw [Submodule.mem_map]\n  use b - v, hshift b hb\n  simp [\u2190 hb']\n  abel\n\n", "theoremStatement": "lemma conclusion_transfers {A B : Set G}\n    (G': AddSubgroup G) (A' B' : Set G')\n    (hA : IsShift A A') (hB : IsShift B B') [Finite A'] [Finite B'] (hA' : A'.Nonempty) (hB' : B'.Nonempty)\n    (h : WeakPFRAsymmConclusion A' B') : WeakPFRAsymmConclusion A B ", "theoremName": "conclusion_transfers", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 986, 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have : Nonempty A' := hA'.to_subtype\n  have : Nonempty B' := hB'.to_subtype\n  rcases h with \u27e8A'', B'', hA'', hB'', hA''_non, hB''_non, hcard_ineq, hdim_ineq\u27e9\n  rcases hA with \u27e8 x, hA \u27e9\n  set f : G' \u2192 G := fun a \u21a6 (a : G) + x\n  have hf : Function.Injective f := by\n    intro y z hyz\n    simp [f] at hyz\n    exact hyz\n  have hA' : A = f '' A' := by\n    simp_rw [hA, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, f, add_comm]; rfl\n  rcases hB with \u27e8 y, hB \u27e9\n  set g : G' \u2192 G := fun a \u21a6 (a : G) + y\n  have hg : Function.Injective g := by\n    intro y z hyz\n    simp [g] at hyz\n    exact hyz\n  have hB' : B = g '' B' := by\n    simp_rw [hB, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, g, add_comm]; rfl\n  use f '' A'', g '' B''\n  have : d\u1d64[A # B] = d\u1d64[A' # B'] := by\n    rw [<-rdist_set_of_inj _ _ (\u03c6 := G'.subtype) Subtype.val_injective, <-rdist_set_add_const (G'.subtype '' A') (G'.subtype '' B') x y]\n    congr\n    . rw [hA]\n      ext y\n      simp [Set.mem_vadd_set]\n      constructor\n      . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n        have : x + z + -x \u2208 G' := by simp [w]\n        use this\n        simp\n        convert hw\n      rintro \u27e8 h, ha \u27e9\n      use y + -x\n      constructor\n      . use h\n      abel\n    rw [hB]\n    ext x\n    simp [Set.mem_vadd_set]\n    constructor\n    . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n      have : y + z + -y \u2208 G' := by simp [w]\n      use this\n      simp\n      convert hw\n    rintro \u27e8 h, ha \u27e9\n    use x + -y\n    constructor\n    . use h\n    abel\n\n\n  refine \u27e8 ?_, ?_, ?_, ?_, ?_, ?_ \u27e9\n  . simp [hA', hf, hA'']\n  . simp [hB', hg, hB'']\n  . simp [hA''_non]\n  . simp [hB''_non]\n  . convert hcard_ineq using 2\n    . congr 3\n      . rw [hA', Nat.card_image_of_injective hf]\n      . rw [hB', Nat.card_image_of_injective hg]\n      . rw [Nat.card_image_of_injective hf]\n      rw [Nat.card_image_of_injective hg]\n  convert LE.le.trans _ hdim_ineq using 2\n  norm_cast\n  apply max_le_max\n  . exact dimension_of_shift A'' x\n  \u00b7 exact dimension_of_shift B'' y", "proofType": "tactic", "proofLengthLines": 69, "proofLengthTokens": 1978}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n  (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n  classical\n  let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n  have : Function.RightInverse g f := by\n    intro n\n    by_cases h : n = 0\n    . simp [g, h, hf]\n    \u00b7 simp [g, h, Function.surjInv_eq hs n]\n  have hg : g 0 = 0 := by simp [g]\n  have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n  intro F\n  use Finsupp.mapRange g hg F\n  rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n  convert Finsupp.mapRange_id F\n  convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n    rintro x hx hn\n    rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n    rcases hn with \u27e8 n, hn, hn' \u27e9\n    apply_fun Module.Free.repr \u2124 G at hn'\n    simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n    linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n.  Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n  let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n  ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  have hH_elem : ElementaryAddCommGroup H 2 := by\n    apply ElementaryAddCommGroup.quotient_group (by decide)\n    intro x; rw [AddMonoidHom.mem_range]\n    use x\n    rw [zsmulAddGroupHom_apply]\n    norm_cast\n  let B := Module.Free.ChooseBasisIndex \u2124 G\n  let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n  have hB_fin : Fintype B := by infer_instance\n  have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n    rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n  have hH_module : Module (ZMod 2) H := by infer_instance\n  let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n  let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n  have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n    intro x hx\n    simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n    simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n      Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n    rcases hx with \u27e8y, rfl\u27e9\n    ext b\n    simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n      Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n    left\n    exact ZMod.nat_cast_self 2\n  let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n  have hsur : Function.Surjective g := by\n    have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n    have h2 := h1.comp bG.repr.surjective\n    have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n      ext x b\n      simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n        Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n      rw [QuotientAddGroup.lift_mk]\n      simp [mod, f]\n    rw [h3] at h2\n    apply Function.Surjective.of_comp h2\n  have hinj : Function.Injective g := by\n    rw [injective_iff_map_eq_zero]\n    intro x hx\n    rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n    simp only [QuotientAddGroup.mk'_apply, g] at hx\n    rw [QuotientAddGroup.lift_mk] at hx\n    simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n      Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n      ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n    replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n    let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n    let z' := (Finsupp.equivFunOnFinite).symm z\n    change \u2200 b : B, 2 * z' b = (f y) b at hx\n    let x' := bG.repr.symm z'\n    rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n    simp_rw [\u03c8, zsmulAddGroupHom_apply]\n    use x'\n    change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n    rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n    rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n    change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n    congr\n    ext b\n    rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n  rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n  have bH : Basis B (ZMod 2) H := by\n    constructor\n    exact {\n      toFun := g\n      invFun := g'\n      left_inv := hg'.1\n      right_inv := hg'.2\n      map_add' := AddMonoidHom.map_add _\n      map_smul' := by\n        intro r x\n        rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n        change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n        rw [intCast_smul, intCast_smul]\n        exact AddMonoidHom.map_zsmul g x n\n    }\n  have hH_fin : Fintype H := Module.fintypeOfFintype bH\n  have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n    rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n    congr\n  exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n  let H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  have h1 : G\u2082 \u2264 N := by\n    intro x hx\n    rw [\u2190 eq_zero_iff] at hx\n    have : \u03c6 x = 0 := hx\n    simp [N, this, AddSubgroup.zero_mem H']\n  set H'' := AddSubgroup.map (mk' G\u2082) N\n  have h2 : H' = H'' := by\n    change H' = AddSubgroup.map (mk' G\u2082) N\n    rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n    simp\n  let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n  let e2 := quotientAddEquivOfEq h2\n  set e := e2.trans e1\n  use e\n  intro x\n  convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n    [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n    z \u2208 A := by\n  contrapose! hz\n  have : Disjoint {z} A := by simp [hz]\n  replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n  simp [hA] at this\n  have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n  simp [this] at h\n  assumption\n\n/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n    \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n    Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n    By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n    (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n      40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n      \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) := by\n  have : Nonempty A := hnA.to_subtype\n  have : Nonempty B := hnB.to_subtype\n  obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  let _mH : MeasurableSpace H := \u22a4\n  have h_fintype : Fintype H := Fintype.ofFinite H\n  have h_torsionfree := torsion_free (G := G)\n\n  obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n  obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n    exists_isUniform_measureSpace' B\n\n  rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n  let N := AddSubgroup.comap \u03c6 H'\n  set \u03c6' := QuotientAddGroup.mk' N\n  have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n  have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n    constructor\n    intro x\n    exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n  rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n  rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n    \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n  have : Nonempty Ax := hnAx.to_subtype\n  have : Nonempty By := hnBy.to_subtype\n  have Axf : Finite Ax := by rw [hAx]; infer_instance\n  have Byf : Finite By := by rw [hBy]; infer_instance\n\n  have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n  have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n  rw [rdist_symm] at h2\n  rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n  -- using explicit .toFun casts as this saves a lot of heartbeats\n  change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n  change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n  replace hH1 : log (Nat.card H') \u2264 40  * d\u1d64[A # B] := by\n    apply hH1.trans\n    linarith\n  replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n    rw [mul_comm, \u2190 log_rpow (by norm_num)]\n    norm_cast\n    classical\n    rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n    all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n  use N, x, y, Ax, By\n  refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n  replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n    set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n    set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n    have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n    have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n    change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n    have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n    have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n    have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n    rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n    have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] :=  rdist_of_inj hX hY e.toAddMonoidHom he_inj\n    rwa [this]\n\n  set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n  set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n  have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n  have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n  rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n  swap\n  . rw [\u2190 h] at hH2\n    have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n    have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n    have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n    have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n    have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n      change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n      set Af := A.toFinite.toFinset\n      have hUAf : IsUniform Af UA := by\n        convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n      have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UA \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n      replace this := single ((\u2119).map UA) hx this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hxx : Ax = A := by\n      have h : hnAx.some \u2208 Ax := hnAx.some_mem\n      simp [hAx] at h \u22a2\n      have := hAAx h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hAAx hz\n\n    have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n      change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n      set Bf := B.toFinite.toFinset\n      have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n      have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UB \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n      replace this := single ((\u2119).map UB) hy this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hyy : By = B := by\n      have h : hnBy.some \u2208 By := hnBy.some_mem\n      simp [hBy] at h \u22a2\n      have := hBBy h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hBBy hz\n\n    simp [hxx, hyy]\n\n  have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n      congr\n      convert (four_logs ?_ ?_ ?_ ?_).symm\n      all_goals norm_cast; exact Nat.card_pos\n    _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n    _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n      apply mul_le_mul_of_nonneg_right hH2\n      have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n      replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n      rw [\u2190 mul_le_mul_left this]\n      apply le_trans _ hcard_ineq\n      rw [mul_zero]\n      change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n      rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n      apply Real.log_nonneg\n      rw [one_le_div]\n      gcongr\n      . apply Nat.card_mono\n        . exact Set.toFinite A\n        rw [hAx]; exact Set.inter_subset_left _ _\n      apply Nat.card_mono\n      . exact Set.toFinite B\n      rw [hBy]; exact Set.inter_subset_left _ _\n      norm_cast\n      exact mul_pos Nat.card_pos Nat.card_pos\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n      rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n  exact (mul_le_mul_left h).mp this\n\n/-- Separating out the conclusion of `weak_PFR_asymm` for convenience of induction arguments.-/\ndef WeakPFRAsymmConclusion (A B : Set G) : Prop :=\n  \u2203 A' B' : Set G, A' \u2286 A \u2227 B' \u2286 B \u2227 A'.Nonempty \u2227 B'.Nonempty \u2227\n  log (((Nat.card A) * (Nat.card B)) / ((Nat.card A') * (Nat.card B'))) \u2264 34 * d\u1d64[A # B] \u2227\n  max (dimension A') (dimension B') \u2264 (40 / log 2) * d\u1d64[A # B]\n\n/-- The property of two sets A,B of a group G not being contained in cosets of the same proper subgroup -/\ndef not_in_coset {G: Type u} [AddCommGroup G] (A B : Set G) : Prop := AddSubgroup.closure ((A-A) \u222a (B-B)) = \u22a4\n\n\n/-- In fact one has equality here, but this is tricker to prove and not needed for the argument. -/\nlemma dimension_of_shift {G: Type u} [AddCommGroup G]\n  {H: AddSubgroup G} (A : Set H) (x : G) :\n  dimension ((fun a:H \u21a6 (a:G) + x) '' A) \u2264 dimension A := by\n  classical\n  rcases Nat.find_spec (exists_coset_cover A) with \u27e8 S, v, hrank, hshift \u27e9\n  change FiniteDimensional.finrank \u2124 S = dimension A at hrank\n  rw [\u2190 hrank]\n  convert dimension_le_of_coset_cover _ (Submodule.map H.subtype.toIntLinearMap S) (x+v) ?_\n  . apply LinearEquiv.finrank_eq\n    exact Submodule.equivMapOfInjective _ (by simpa using Subtype.val_injective) _\n  intro a ha\n  rw [Set.mem_image] at ha\n  rcases ha with \u27e8 b, \u27e8 hb, hb'\u27e9 \u27e9\n  rw [Submodule.mem_map]\n  use b - v, hshift b hb\n  simp [\u2190 hb']\n  abel\n\nlemma conclusion_transfers {A B : Set G}\n    (G': AddSubgroup G) (A' B' : Set G')\n    (hA : IsShift A A') (hB : IsShift B B') [Finite A'] [Finite B'] (hA' : A'.Nonempty) (hB' : B'.Nonempty)\n    (h : WeakPFRAsymmConclusion A' B') : WeakPFRAsymmConclusion A B := by\n  have : Nonempty A' := hA'.to_subtype\n  have : Nonempty B' := hB'.to_subtype\n  rcases h with \u27e8A'', B'', hA'', hB'', hA''_non, hB''_non, hcard_ineq, hdim_ineq\u27e9\n  rcases hA with \u27e8 x, hA \u27e9\n  set f : G' \u2192 G := fun a \u21a6 (a : G) + x\n  have hf : Function.Injective f := by\n    intro y z hyz\n    simp [f] at hyz\n    exact hyz\n  have hA' : A = f '' A' := by\n    simp_rw [hA, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, f, add_comm]; rfl\n  rcases hB with \u27e8 y, hB \u27e9\n  set g : G' \u2192 G := fun a \u21a6 (a : G) + y\n  have hg : Function.Injective g := by\n    intro y z hyz\n    simp [g] at hyz\n    exact hyz\n  have hB' : B = g '' B' := by\n    simp_rw [hB, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, g, add_comm]; rfl\n  use f '' A'', g '' B''\n  have : d\u1d64[A # B] = d\u1d64[A' # B'] := by\n    rw [<-rdist_set_of_inj _ _ (\u03c6 := G'.subtype) Subtype.val_injective, <-rdist_set_add_const (G'.subtype '' A') (G'.subtype '' B') x y]\n    congr\n    . rw [hA]\n      ext y\n      simp [Set.mem_vadd_set]\n      constructor\n      . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n        have : x + z + -x \u2208 G' := by simp [w]\n        use this\n        simp\n        convert hw\n      rintro \u27e8 h, ha \u27e9\n      use y + -x\n      constructor\n      . use h\n      abel\n    rw [hB]\n    ext x\n    simp [Set.mem_vadd_set]\n    constructor\n    . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n      have : y + z + -y \u2208 G' := by simp [w]\n      use this\n      simp\n      convert hw\n    rintro \u27e8 h, ha \u27e9\n    use x + -y\n    constructor\n    . use h\n    abel\n\n\n  refine \u27e8 ?_, ?_, ?_, ?_, ?_, ?_ \u27e9\n  . simp [hA', hf, hA'']\n  . simp [hB', hg, hB'']\n  . simp [hA''_non]\n  . simp [hB''_non]\n  . convert hcard_ineq using 2\n    . congr 3\n      . rw [hA', Nat.card_image_of_injective hf]\n      . rw [hB', Nat.card_image_of_injective hg]\n      . rw [Nat.card_image_of_injective hf]\n      rw [Nat.card_image_of_injective hg]\n  convert LE.le.trans _ hdim_ineq using 2\n  norm_cast\n  apply max_le_max\n  . exact dimension_of_shift A'' x\n  \u00b7 exact dimension_of_shift B'' y\n\n", "theoremStatement": "/-- If $A,B\\subseteq \\mathbb{Z}^d$ are finite non-empty sets then there exist non-empty $A'\\subseteq A$ and $B'\\subseteq B$ such that\n\\[\\log\\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A'\\rvert\\lvert B'\\rvert}\\leq 34 d[U_A;U_B]\\]\nsuch that $\\max(\\dim A',\\dim B')\\leq \\frac{40}{\\log 2} d[U_A;U_B]$. -/\nlemma weak_PFR_asymm (A B : Set G) [Finite A] [Finite B] (hA : A.Nonempty) (hB : B.Nonempty) : WeakPFRAsymmConclusion A B  ", "theoremName": "weak_PFR_asymm", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 1060, "tokenPositionInFile": 54134, "theoremPositionInFile": 28}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 10, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 442, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  let P : \u2115 \u2192 Prop := fun M \u21a6 (\u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G) (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G) (_hG_sing: MeasurableSingletonClass G) (A B: Set G) (_hA_fin: Finite A) (_hB_fin: Finite B) (_hA_non: A.Nonempty) (_hB_non: B.Nonempty) (_hM : (Nat.card A) + (Nat.card B) \u2264 M), WeakPFRAsymmConclusion A B)\n  suffices \u2200 M, (\u2200 M', M' < M \u2192 P M') \u2192 P M by\n    set M := (Nat.card A) + (Nat.card B)\n    have hM : (Nat.card A) + (Nat.card B) \u2264 M := Nat.le_refl _\n    convert (Nat.strong_induction_on (p := P) M this) G \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a _ \u2039_\u203a A B \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a hM\n  intro M h_induct\n  -- wlog we can assume A, B are not in cosets of a smaller subgroup\n  suffices \u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G)\n    (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G)\n    (_hG_sing : MeasurableSingletonClass G) (A B : Set G) (_hA_fin : Finite A) (_hB_fin : Finite B) (_hA_non : A.Nonempty) (_hB_non : B.Nonempty) (_hM : Nat.card A + Nat.card B \u2264 M)\n    (_hnot : NotInCoset A B), WeakPFRAsymmConclusion A B by\n    intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM\n    obtain \u27e8 G', A', B', hAA', hBB', hnot' \u27e9 := wlog_notInCoset hA_non hB_non\n    have hG'_fin : Module.Finite \u2124 G' :=\n      Module.Finite.iff_fg (N := AddSubgroup.toIntSubmodule G').2 (IsNoetherian.noetherian _)\n\n    have hG'_free : Module.Free \u2124 G' := by\n      rcases Submodule.nonempty_basis_of_pid (Module.Free.chooseBasis \u2124 G) (AddSubgroup.toIntSubmodule G') with \u27e8 n, \u27e8 b \u27e9 \u27e9\n      exact Module.Free.of_basis b\n    have hAA'_card: Nat.card A = Nat.card A' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hAA'.card_congr\n    have hBB'_card: Nat.card B = Nat.card B' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hBB'.card_congr\n    have hA_non' : Nonempty A := Set.nonempty_coe_sort.mpr hA_non\n    have hB_non' : Nonempty B := Set.nonempty_coe_sort.mpr hB_non\n\n    rw [hAA'_card, hBB'_card] at hM\n\n    have hA'_nonfin : A'.Nonempty \u2227 Finite A' := by\n      have := Nat.card_pos (\u03b1 := A)\n      rw [hAA'_card, Nat.card_pos_iff] at this\n      exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n    have hB'_nonfin : B'.Nonempty \u2227 Finite B' := by\n      have := Nat.card_pos (\u03b1 := B)\n      rw [hBB'_card, Nat.card_pos_iff] at this\n      exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n    obtain \u27e8 hA'_non, hA'_fin \u27e9 := hA'_nonfin\n    obtain \u27e8 hB'_non, hB'_fin \u27e9 := hB'_nonfin\n\n    replace this := this G' _ hG'_free hG'_fin (by infer_instance) (by infer_instance) (by infer_instance) A' B' hA'_fin hB'_fin hA'_non hB'_non hM hnot'\n    exact conclusion_transfers G' A' B' hAA' hBB' hA'_non hB'_non this\n  intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM hnot\n  rcases weak_PFR_asymm_prelim A B hA_non hB_non with \u27e8 N, x, y, Ax, By, hAx_non, hBy_non, hAx_fin, hBy_fin, hAx, hBy, hdim, hcard\u27e9\n  have hAxA : Ax \u2286 A := by rw [hAx]; simp\n  have hByB : By \u2286 B := by rw [hBy]; simp\n  have hA_pos : (0 : \u211d) < Nat.card A := Nat.cast_pos.mpr (@Nat.card_pos _ hA_non.to_subtype _)\n  have hB_pos : (0 : \u211d) < Nat.card B := Nat.cast_pos.mpr (@Nat.card_pos _ hB_non.to_subtype _)\n\n  rcases lt_or_ge (Nat.card Ax + Nat.card By) (Nat.card A + Nat.card B) with h | h\n  . replace h := h_induct (Nat.card Ax + Nat.card By) (h.trans_le hM) G hG_comm hG_free hG_fin hG_count hG_mes hG_sing Ax By (Set.finite_coe_iff.mpr hAx_fin) (Set.finite_coe_iff.mpr hBy_fin) hAx_non hBy_non (Eq.le rfl)\n    rcases h with \u27e8 A', B', hA', hB', hA'_non, hB'_non, hcard_ineq, hdim_ineq \u27e9\n    use A', B'\n    have hAx_fin' := Set.finite_coe_iff.mpr hAx_fin\n    have hBy_fin' := Set.finite_coe_iff.mpr hBy_fin\n    have hA'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hAx_fin hA')\n    have hB'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hBy_fin hB')\n    have hAx_non' := Set.nonempty_coe_sort.mpr hAx_non\n    have hBy_non' := Set.nonempty_coe_sort.mpr hBy_non\n    have hA'_non' := Set.nonempty_coe_sort.mpr hA'_non\n    have hB'_non' := Set.nonempty_coe_sort.mpr hB'_non\n    have hAx_pos : (0 : \u211d) < Nat.card Ax := Nat.cast_pos.mpr Nat.card_pos\n    have hBy_pos : (0 : \u211d) < Nat.card By := Nat.cast_pos.mpr Nat.card_pos\n    have hA'_pos : (0 : \u211d) < Nat.card A' := Nat.cast_pos.mpr Nat.card_pos\n    have hB'_pos : (0 : \u211d) < Nat.card B' := Nat.cast_pos.mpr Nat.card_pos\n    have hAxA_le : (Nat.card Ax : \u211d) \u2264 (Nat.card A : \u211d) := Nat.cast_le.mpr (Nat.card_mono A.toFinite hAxA)\n    have hByB_le : (Nat.card By : \u211d) \u2264 (Nat.card B : \u211d) := Nat.cast_le.mpr (Nat.card_mono B.toFinite hByB)\n\n    refine \u27e8 hA'.trans hAxA, hB'.trans hByB, hA'_non, hB'_non, ?_, ?_ \u27e9\n    . rw [four_logs hA_pos hB_pos hA'_pos hB'_pos]\n      rw [four_logs hAx_pos hBy_pos hA'_pos hB'_pos] at hcard_ineq\n      linarith only [hcard, hcard_ineq]\n    apply hdim_ineq.trans\n    gcongr\n    linarith only [Real.log_le_log hAx_pos hAxA_le, Real.log_le_log hBy_pos hByB_le, hcard]\n  use A, B\n  refine \u27e8 Eq.subset rfl, Eq.subset rfl, hA_non, hB_non, ?_, ?_ \u27e9\n  . have := hA_non.to_subtype\n    have := hB_non.to_subtype\n    apply LE.le.trans _ <| mul_nonneg (by norm_num) <| rdist_set_nonneg A B\n    rw [div_self (by positivity)]\n    simp\n  have hAx_eq : Ax = A := by\n    apply Set.Finite.eq_of_subset_of_card_le A.toFinite hAxA\n    linarith only [h, Nat.card_mono B.toFinite hByB]\n  have hBy_eq : By = B := by\n    apply Set.Finite.eq_of_subset_of_card_le B.toFinite hByB\n    linarith only [h, Nat.card_mono A.toFinite hAxA]\n  have hN : N = \u22a4 := by\n    have : (A-A) \u222a (B-B) \u2286 N := by\n      rw [\u2190 hAx_eq, \u2190 hBy_eq, hAx, hBy]\n      intro z hz\n      simp [Set.mem_sub] at hz\n      convert (QuotientAddGroup.eq_zero_iff z).mp ?_\n      . infer_instance\n      rcases hz with \u27e8 a, \u27e8 -, ha\u27e9, a', \u27e8-, ha'\u27e9, haa' \u27e9 | \u27e8 b, \u27e8 -, hb\u27e9, b', \u27e8 -,hb'\u27e9, hbb' \u27e9\n      . rw [\u2190 haa']; simp [ha, ha']\n      rw [\u2190 hbb']; simp [hb, hb']\n    rw [\u2190 AddSubgroup.closure_le, hnot] at this\n    exact top_le_iff.mp this\n  have : Nat.card (G \u29f8 N) = 1 := by\n    rw [Nat.card_eq_one_iff_unique]\n    constructor\n    . rw [hN]\n      exact  QuotientAddGroup.subsingleton_quotient_top\n    infer_instance\n  simp [this] at hdim\n  rw [\u2190 le_div_iff' (by positivity)] at hdim\n  convert LE.le.trans ?_ hdim using 1\n  . field_simp\n  simp\n  exact \u27e8 dimension_le_rank A, dimension_le_rank B \u27e9", "proofType": "tactic", "proofLengthLines": 109, "proofLengthTokens": 6396}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n  (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n  classical\n  let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n  have : Function.RightInverse g f := by\n    intro n\n    by_cases h : n = 0\n    . simp [g, h, hf]\n    \u00b7 simp [g, h, Function.surjInv_eq hs n]\n  have hg : g 0 = 0 := by simp [g]\n  have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n  intro F\n  use Finsupp.mapRange g hg F\n  rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n  convert Finsupp.mapRange_id F\n  convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n    rintro x hx hn\n    rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n    rcases hn with \u27e8 n, hn, hn' \u27e9\n    apply_fun Module.Free.repr \u2124 G at hn'\n    simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n    linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n.  Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n  let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n  ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  have hH_elem : ElementaryAddCommGroup H 2 := by\n    apply ElementaryAddCommGroup.quotient_group (by decide)\n    intro x; rw [AddMonoidHom.mem_range]\n    use x\n    rw [zsmulAddGroupHom_apply]\n    norm_cast\n  let B := Module.Free.ChooseBasisIndex \u2124 G\n  let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n  have hB_fin : Fintype B := by infer_instance\n  have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n    rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n  have hH_module : Module (ZMod 2) H := by infer_instance\n  let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n  let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n  have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n    intro x hx\n    simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n    simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n      Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n    rcases hx with \u27e8y, rfl\u27e9\n    ext b\n    simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n      Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n    left\n    exact ZMod.nat_cast_self 2\n  let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n  have hsur : Function.Surjective g := by\n    have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n    have h2 := h1.comp bG.repr.surjective\n    have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n      ext x b\n      simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n        Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n      rw [QuotientAddGroup.lift_mk]\n      simp [mod, f]\n    rw [h3] at h2\n    apply Function.Surjective.of_comp h2\n  have hinj : Function.Injective g := by\n    rw [injective_iff_map_eq_zero]\n    intro x hx\n    rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n    simp only [QuotientAddGroup.mk'_apply, g] at hx\n    rw [QuotientAddGroup.lift_mk] at hx\n    simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n      Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n      ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n    replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n    let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n    let z' := (Finsupp.equivFunOnFinite).symm z\n    change \u2200 b : B, 2 * z' b = (f y) b at hx\n    let x' := bG.repr.symm z'\n    rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n    simp_rw [\u03c8, zsmulAddGroupHom_apply]\n    use x'\n    change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n    rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n    rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n    change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n    congr\n    ext b\n    rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n  rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n  have bH : Basis B (ZMod 2) H := by\n    constructor\n    exact {\n      toFun := g\n      invFun := g'\n      left_inv := hg'.1\n      right_inv := hg'.2\n      map_add' := AddMonoidHom.map_add _\n      map_smul' := by\n        intro r x\n        rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n        change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n        rw [intCast_smul, intCast_smul]\n        exact AddMonoidHom.map_zsmul g x n\n    }\n  have hH_fin : Fintype H := Module.fintypeOfFintype bH\n  have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n    rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n    congr\n  exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n  let H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  have h1 : G\u2082 \u2264 N := by\n    intro x hx\n    rw [\u2190 eq_zero_iff] at hx\n    have : \u03c6 x = 0 := hx\n    simp [N, this, AddSubgroup.zero_mem H']\n  set H'' := AddSubgroup.map (mk' G\u2082) N\n  have h2 : H' = H'' := by\n    change H' = AddSubgroup.map (mk' G\u2082) N\n    rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n    simp\n  let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n  let e2 := quotientAddEquivOfEq h2\n  set e := e2.trans e1\n  use e\n  intro x\n  convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n    [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n    z \u2208 A := by\n  contrapose! hz\n  have : Disjoint {z} A := by simp [hz]\n  replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n  simp [hA] at this\n  have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n  simp [this] at h\n  assumption\n\n/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n    \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n    Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n    By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n    (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n      40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n      \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) := by\n  have : Nonempty A := hnA.to_subtype\n  have : Nonempty B := hnB.to_subtype\n  obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  let _mH : MeasurableSpace H := \u22a4\n  have h_fintype : Fintype H := Fintype.ofFinite H\n  have h_torsionfree := torsion_free (G := G)\n\n  obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n  obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n    exists_isUniform_measureSpace' B\n\n  rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n  let N := AddSubgroup.comap \u03c6 H'\n  set \u03c6' := QuotientAddGroup.mk' N\n  have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n  have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n    constructor\n    intro x\n    exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n  rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n  rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n    \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n  have : Nonempty Ax := hnAx.to_subtype\n  have : Nonempty By := hnBy.to_subtype\n  have Axf : Finite Ax := by rw [hAx]; infer_instance\n  have Byf : Finite By := by rw [hBy]; infer_instance\n\n  have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n  have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n  rw [rdist_symm] at h2\n  rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n  -- using explicit .toFun casts as this saves a lot of heartbeats\n  change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n  change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n  replace hH1 : log (Nat.card H') \u2264 40  * d\u1d64[A # B] := by\n    apply hH1.trans\n    linarith\n  replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n    rw [mul_comm, \u2190 log_rpow (by norm_num)]\n    norm_cast\n    classical\n    rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n    all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n  use N, x, y, Ax, By\n  refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n  replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n    set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n    set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n    have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n    have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n    change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n    have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n    have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n    have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n    rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n    have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] :=  rdist_of_inj hX hY e.toAddMonoidHom he_inj\n    rwa [this]\n\n  set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n  set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n  have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n  have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n  rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n  swap\n  . rw [\u2190 h] at hH2\n    have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n    have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n    have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n    have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n    have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n      change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n      set Af := A.toFinite.toFinset\n      have hUAf : IsUniform Af UA := by\n        convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n      have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UA \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n      replace this := single ((\u2119).map UA) hx this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hxx : Ax = A := by\n      have h : hnAx.some \u2208 Ax := hnAx.some_mem\n      simp [hAx] at h \u22a2\n      have := hAAx h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hAAx hz\n\n    have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n      change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n      set Bf := B.toFinite.toFinset\n      have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n      have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UB \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n      replace this := single ((\u2119).map UB) hy this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hyy : By = B := by\n      have h : hnBy.some \u2208 By := hnBy.some_mem\n      simp [hBy] at h \u22a2\n      have := hBBy h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hBBy hz\n\n    simp [hxx, hyy]\n\n  have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n      congr\n      convert (four_logs ?_ ?_ ?_ ?_).symm\n      all_goals norm_cast; exact Nat.card_pos\n    _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n    _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n      apply mul_le_mul_of_nonneg_right hH2\n      have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n      replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n      rw [\u2190 mul_le_mul_left this]\n      apply le_trans _ hcard_ineq\n      rw [mul_zero]\n      change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n      rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n      apply Real.log_nonneg\n      rw [one_le_div]\n      gcongr\n      . apply Nat.card_mono\n        . exact Set.toFinite A\n        rw [hAx]; exact Set.inter_subset_left _ _\n      apply Nat.card_mono\n      . exact Set.toFinite B\n      rw [hBy]; exact Set.inter_subset_left _ _\n      norm_cast\n      exact mul_pos Nat.card_pos Nat.card_pos\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n      rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n  exact (mul_le_mul_left h).mp this\n\n/-- Separating out the conclusion of `weak_PFR_asymm` for convenience of induction arguments.-/\ndef WeakPFRAsymmConclusion (A B : Set G) : Prop :=\n  \u2203 A' B' : Set G, A' \u2286 A \u2227 B' \u2286 B \u2227 A'.Nonempty \u2227 B'.Nonempty \u2227\n  log (((Nat.card A) * (Nat.card B)) / ((Nat.card A') * (Nat.card B'))) \u2264 34 * d\u1d64[A # B] \u2227\n  max (dimension A') (dimension B') \u2264 (40 / log 2) * d\u1d64[A # B]\n\n/-- The property of two sets A,B of a group G not being contained in cosets of the same proper subgroup -/\ndef not_in_coset {G: Type u} [AddCommGroup G] (A B : Set G) : Prop := AddSubgroup.closure ((A-A) \u222a (B-B)) = \u22a4\n\n\n/-- In fact one has equality here, but this is tricker to prove and not needed for the argument. -/\nlemma dimension_of_shift {G: Type u} [AddCommGroup G]\n  {H: AddSubgroup G} (A : Set H) (x : G) :\n  dimension ((fun a:H \u21a6 (a:G) + x) '' A) \u2264 dimension A := by\n  classical\n  rcases Nat.find_spec (exists_coset_cover A) with \u27e8 S, v, hrank, hshift \u27e9\n  change FiniteDimensional.finrank \u2124 S = dimension A at hrank\n  rw [\u2190 hrank]\n  convert dimension_le_of_coset_cover _ (Submodule.map H.subtype.toIntLinearMap S) (x+v) ?_\n  . apply LinearEquiv.finrank_eq\n    exact Submodule.equivMapOfInjective _ (by simpa using Subtype.val_injective) _\n  intro a ha\n  rw [Set.mem_image] at ha\n  rcases ha with \u27e8 b, \u27e8 hb, hb'\u27e9 \u27e9\n  rw [Submodule.mem_map]\n  use b - v, hshift b hb\n  simp [\u2190 hb']\n  abel\n\nlemma conclusion_transfers {A B : Set G}\n    (G': AddSubgroup G) (A' B' : Set G')\n    (hA : IsShift A A') (hB : IsShift B B') [Finite A'] [Finite B'] (hA' : A'.Nonempty) (hB' : B'.Nonempty)\n    (h : WeakPFRAsymmConclusion A' B') : WeakPFRAsymmConclusion A B := by\n  have : Nonempty A' := hA'.to_subtype\n  have : Nonempty B' := hB'.to_subtype\n  rcases h with \u27e8A'', B'', hA'', hB'', hA''_non, hB''_non, hcard_ineq, hdim_ineq\u27e9\n  rcases hA with \u27e8 x, hA \u27e9\n  set f : G' \u2192 G := fun a \u21a6 (a : G) + x\n  have hf : Function.Injective f := by\n    intro y z hyz\n    simp [f] at hyz\n    exact hyz\n  have hA' : A = f '' A' := by\n    simp_rw [hA, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, f, add_comm]; rfl\n  rcases hB with \u27e8 y, hB \u27e9\n  set g : G' \u2192 G := fun a \u21a6 (a : G) + y\n  have hg : Function.Injective g := by\n    intro y z hyz\n    simp [g] at hyz\n    exact hyz\n  have hB' : B = g '' B' := by\n    simp_rw [hB, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, g, add_comm]; rfl\n  use f '' A'', g '' B''\n  have : d\u1d64[A # B] = d\u1d64[A' # B'] := by\n    rw [<-rdist_set_of_inj _ _ (\u03c6 := G'.subtype) Subtype.val_injective, <-rdist_set_add_const (G'.subtype '' A') (G'.subtype '' B') x y]\n    congr\n    . rw [hA]\n      ext y\n      simp [Set.mem_vadd_set]\n      constructor\n      . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n        have : x + z + -x \u2208 G' := by simp [w]\n        use this\n        simp\n        convert hw\n      rintro \u27e8 h, ha \u27e9\n      use y + -x\n      constructor\n      . use h\n      abel\n    rw [hB]\n    ext x\n    simp [Set.mem_vadd_set]\n    constructor\n    . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n      have : y + z + -y \u2208 G' := by simp [w]\n      use this\n      simp\n      convert hw\n    rintro \u27e8 h, ha \u27e9\n    use x + -y\n    constructor\n    . use h\n    abel\n\n\n  refine \u27e8 ?_, ?_, ?_, ?_, ?_, ?_ \u27e9\n  . simp [hA', hf, hA'']\n  . simp [hB', hg, hB'']\n  . simp [hA''_non]\n  . simp [hB''_non]\n  . convert hcard_ineq using 2\n    . congr 3\n      . rw [hA', Nat.card_image_of_injective hf]\n      . rw [hB', Nat.card_image_of_injective hg]\n      . rw [Nat.card_image_of_injective hf]\n      rw [Nat.card_image_of_injective hg]\n  convert LE.le.trans _ hdim_ineq using 2\n  norm_cast\n  apply max_le_max\n  . exact dimension_of_shift A'' x\n  \u00b7 exact dimension_of_shift B'' y\n\n/-- If $A,B\\subseteq \\mathbb{Z}^d$ are finite non-empty sets then there exist non-empty $A'\\subseteq A$ and $B'\\subseteq B$ such that\n\\[\\log\\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A'\\rvert\\lvert B'\\rvert}\\leq 34 d[U_A;U_B]\\]\nsuch that $\\max(\\dim A',\\dim B')\\leq \\frac{40}{\\log 2} d[U_A;U_B]$. -/\nlemma weak_PFR_asymm (A B : Set G) [Finite A] [Finite B] (hA : A.Nonempty) (hB : B.Nonempty) : WeakPFRAsymmConclusion A B  := by\n  let P : \u2115 \u2192 Prop := fun M \u21a6 (\u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G) (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G) (_hG_sing: MeasurableSingletonClass G) (A B: Set G) (_hA_fin: Finite A) (_hB_fin: Finite B) (_hA_non: A.Nonempty) (_hB_non: B.Nonempty) (_hM : (Nat.card A) + (Nat.card B) \u2264 M), WeakPFRAsymmConclusion A B)\n  suffices \u2200 M, (\u2200 M', M' < M \u2192 P M') \u2192 P M by\n    set M := (Nat.card A) + (Nat.card B)\n    have hM : (Nat.card A) + (Nat.card B) \u2264 M := Nat.le_refl _\n    convert (Nat.strong_induction_on (p := P) M this) G \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a _ \u2039_\u203a A B \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a hM\n  intro M h_induct\n  -- wlog we can assume A, B are not in cosets of a smaller subgroup\n  suffices \u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G)\n    (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G)\n    (_hG_sing : MeasurableSingletonClass G) (A B : Set G) (_hA_fin : Finite A) (_hB_fin : Finite B) (_hA_non : A.Nonempty) (_hB_non : B.Nonempty) (_hM : Nat.card A + Nat.card B \u2264 M)\n    (_hnot : NotInCoset A B), WeakPFRAsymmConclusion A B by\n    intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM\n    obtain \u27e8 G', A', B', hAA', hBB', hnot' \u27e9 := wlog_notInCoset hA_non hB_non\n    have hG'_fin : Module.Finite \u2124 G' :=\n      Module.Finite.iff_fg (N := AddSubgroup.toIntSubmodule G').2 (IsNoetherian.noetherian _)\n\n    have hG'_free : Module.Free \u2124 G' := by\n      rcases Submodule.nonempty_basis_of_pid (Module.Free.chooseBasis \u2124 G) (AddSubgroup.toIntSubmodule G') with \u27e8 n, \u27e8 b \u27e9 \u27e9\n      exact Module.Free.of_basis b\n    have hAA'_card: Nat.card A = Nat.card A' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hAA'.card_congr\n    have hBB'_card: Nat.card B = Nat.card B' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hBB'.card_congr\n    have hA_non' : Nonempty A := Set.nonempty_coe_sort.mpr hA_non\n    have hB_non' : Nonempty B := Set.nonempty_coe_sort.mpr hB_non\n\n    rw [hAA'_card, hBB'_card] at hM\n\n    have hA'_nonfin : A'.Nonempty \u2227 Finite A' := by\n      have := Nat.card_pos (\u03b1 := A)\n      rw [hAA'_card, Nat.card_pos_iff] at this\n      exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n    have hB'_nonfin : B'.Nonempty \u2227 Finite B' := by\n      have := Nat.card_pos (\u03b1 := B)\n      rw [hBB'_card, Nat.card_pos_iff] at this\n      exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n    obtain \u27e8 hA'_non, hA'_fin \u27e9 := hA'_nonfin\n    obtain \u27e8 hB'_non, hB'_fin \u27e9 := hB'_nonfin\n\n    replace this := this G' _ hG'_free hG'_fin (by infer_instance) (by infer_instance) (by infer_instance) A' B' hA'_fin hB'_fin hA'_non hB'_non hM hnot'\n    exact conclusion_transfers G' A' B' hAA' hBB' hA'_non hB'_non this\n  intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM hnot\n  rcases weak_PFR_asymm_prelim A B hA_non hB_non with \u27e8 N, x, y, Ax, By, hAx_non, hBy_non, hAx_fin, hBy_fin, hAx, hBy, hdim, hcard\u27e9\n  have hAxA : Ax \u2286 A := by rw [hAx]; simp\n  have hByB : By \u2286 B := by rw [hBy]; simp\n  have hA_pos : (0 : \u211d) < Nat.card A := Nat.cast_pos.mpr (@Nat.card_pos _ hA_non.to_subtype _)\n  have hB_pos : (0 : \u211d) < Nat.card B := Nat.cast_pos.mpr (@Nat.card_pos _ hB_non.to_subtype _)\n\n  rcases lt_or_ge (Nat.card Ax + Nat.card By) (Nat.card A + Nat.card B) with h | h\n  . replace h := h_induct (Nat.card Ax + Nat.card By) (h.trans_le hM) G hG_comm hG_free hG_fin hG_count hG_mes hG_sing Ax By (Set.finite_coe_iff.mpr hAx_fin) (Set.finite_coe_iff.mpr hBy_fin) hAx_non hBy_non (Eq.le rfl)\n    rcases h with \u27e8 A', B', hA', hB', hA'_non, hB'_non, hcard_ineq, hdim_ineq \u27e9\n    use A', B'\n    have hAx_fin' := Set.finite_coe_iff.mpr hAx_fin\n    have hBy_fin' := Set.finite_coe_iff.mpr hBy_fin\n    have hA'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hAx_fin hA')\n    have hB'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hBy_fin hB')\n    have hAx_non' := Set.nonempty_coe_sort.mpr hAx_non\n    have hBy_non' := Set.nonempty_coe_sort.mpr hBy_non\n    have hA'_non' := Set.nonempty_coe_sort.mpr hA'_non\n    have hB'_non' := Set.nonempty_coe_sort.mpr hB'_non\n    have hAx_pos : (0 : \u211d) < Nat.card Ax := Nat.cast_pos.mpr Nat.card_pos\n    have hBy_pos : (0 : \u211d) < Nat.card By := Nat.cast_pos.mpr Nat.card_pos\n    have hA'_pos : (0 : \u211d) < Nat.card A' := Nat.cast_pos.mpr Nat.card_pos\n    have hB'_pos : (0 : \u211d) < Nat.card B' := Nat.cast_pos.mpr Nat.card_pos\n    have hAxA_le : (Nat.card Ax : \u211d) \u2264 (Nat.card A : \u211d) := Nat.cast_le.mpr (Nat.card_mono A.toFinite hAxA)\n    have hByB_le : (Nat.card By : \u211d) \u2264 (Nat.card B : \u211d) := Nat.cast_le.mpr (Nat.card_mono B.toFinite hByB)\n\n    refine \u27e8 hA'.trans hAxA, hB'.trans hByB, hA'_non, hB'_non, ?_, ?_ \u27e9\n    . rw [four_logs hA_pos hB_pos hA'_pos hB'_pos]\n      rw [four_logs hAx_pos hBy_pos hA'_pos hB'_pos] at hcard_ineq\n      linarith only [hcard, hcard_ineq]\n    apply hdim_ineq.trans\n    gcongr\n    linarith only [Real.log_le_log hAx_pos hAxA_le, Real.log_le_log hBy_pos hByB_le, hcard]\n  use A, B\n  refine \u27e8 Eq.subset rfl, Eq.subset rfl, hA_non, hB_non, ?_, ?_ \u27e9\n  . have := hA_non.to_subtype\n    have := hB_non.to_subtype\n    apply LE.le.trans _ <| mul_nonneg (by norm_num) <| rdist_set_nonneg A B\n    rw [div_self (by positivity)]\n    simp\n  have hAx_eq : Ax = A := by\n    apply Set.Finite.eq_of_subset_of_card_le A.toFinite hAxA\n    linarith only [h, Nat.card_mono B.toFinite hByB]\n  have hBy_eq : By = B := by\n    apply Set.Finite.eq_of_subset_of_card_le B.toFinite hByB\n    linarith only [h, Nat.card_mono A.toFinite hAxA]\n  have hN : N = \u22a4 := by\n    have : (A-A) \u222a (B-B) \u2286 N := by\n      rw [\u2190 hAx_eq, \u2190 hBy_eq, hAx, hBy]\n      intro z hz\n      simp [Set.mem_sub] at hz\n      convert (QuotientAddGroup.eq_zero_iff z).mp ?_\n      . infer_instance\n      rcases hz with \u27e8 a, \u27e8 -, ha\u27e9, a', \u27e8-, ha'\u27e9, haa' \u27e9 | \u27e8 b, \u27e8 -, hb\u27e9, b', \u27e8 -,hb'\u27e9, hbb' \u27e9\n      . rw [\u2190 haa']; simp [ha, ha']\n      rw [\u2190 hbb']; simp [hb, hb']\n    rw [\u2190 AddSubgroup.closure_le, hnot] at this\n    exact top_le_iff.mp this\n  have : Nat.card (G \u29f8 N) = 1 := by\n    rw [Nat.card_eq_one_iff_unique]\n    constructor\n    . rw [hN]\n      exact  QuotientAddGroup.subsingleton_quotient_top\n    infer_instance\n  simp [this] at hdim\n  rw [\u2190 le_div_iff' (by positivity)] at hdim\n  convert LE.le.trans ?_ hdim using 1\n  . field_simp\n  simp\n  exact \u27e8 dimension_le_rank A, dimension_le_rank B \u27e9\n\n", "theoremStatement": "/-- If $A\\subseteq \\mathbb{Z}^d$ is a finite non-empty set with $d[U_A;U_A]\\leq \\log K$ then there exists a non-empty $A'\\subseteq A$ such that\n$\\lvert A'\\rvert\\geq K^{-17}\\lvert A\\rvert$\nand $\\dim A'\\leq \\frac{40}{\\log 2} \\log K$. -/\nlemma weak_PFR {A : Set G} [Finite A] {K : \u211d} (hA : A.Nonempty) (hK: 0 < K) (hdist: d\u1d64[A # A] \u2264 log K):\n    \u2203 A' : Set G, A' \u2286 A \u2227 (Nat.card A') \u2265 K^(-17 : \u211d) * (Nat.card A)\n    \u2227 (dimension A') \u2264 (40 / log 2) * log K ", "theoremName": "weak_PFR", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 1174, "tokenPositionInFile": 60954, "theoremPositionInFile": 29}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 3, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 352, "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", 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"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", "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.Util.CompileInductive", "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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rcases weak_PFR_asymm A A hA hA with \u27e8A', A'', hA', hA'', hA'nonempty, hA''nonempty, hcard, hdim\u27e9\n\n  have : \u2203 B : Set G, B \u2286 A \u2227 (Nat.card B) \u2265 (Nat.card A') \u2227 (Nat.card B) \u2265 (Nat.card A'') \u2227 (dimension B) \u2264\nmax (dimension A') (dimension A'') := by\n    rcases lt_or_ge (Nat.card A') (Nat.card A'') with h | h\n    . use A''\n      exact \u27e8hA'', by linarith, by linarith, le_max_right _ _\u27e9\n    use A'\n    exact \u27e8hA', by linarith, by linarith, le_max_left _ _\u27e9\n\n  rcases this with \u27e8B, hB, hBcard, hBcard', hBdim\u27e9\n  use B\n  have hApos : Nat.card A > 0 := by\n    rw [gt_iff_lt, Nat.card_pos_iff]\n    exact \u27e8hA.to_subtype, inferInstance\u27e9\n  have hA'pos : Nat.card A' > 0 := by\n    rw [gt_iff_lt, Nat.card_pos_iff]\n    refine \u27e8 hA'nonempty.to_subtype, Finite.Set.subset _ hA' \u27e9\n  have hA''pos : Nat.card A'' > 0 := by\n    rw [gt_iff_lt, Nat.card_pos_iff]\n    refine \u27e8 hA''nonempty.to_subtype, Finite.Set.subset _ hA'' \u27e9\n  have hBpos : Nat.card B > 0 := by linarith\n\n  refine \u27e8hB, ?_, ?_\u27e9\n  . have := calc 2 * log ((Nat.card A) / (Nat.card B))\n      _ = log ( ((Nat.card A) * (Nat.card A)) / ((Nat.card B) * (Nat.card B)) ) := by\n        convert (log_pow (((Nat.card A) : \u211d)/(Nat.card B)) 2).symm\n        field_simp\n        rw [\u2190 pow_two, \u2190 pow_two]\n      _ \u2264 log ( ((Nat.card A) * (Nat.card A)) / ((Nat.card A') * (Nat.card A'')) ) := by\n        apply log_le_log\n        . positivity\n        gcongr\n      _ \u2264 34 * d\u1d64[A # A] := hcard\n      _ \u2264 34 * log K := mul_le_mul_of_nonneg_left hdist (by linarith)\n      _ = 2 * (17 * log K) := by ring\n      _ = 2 * log (K^17) := by\n        congr\n        convert (log_pow K 17).symm\n    rw [mul_le_mul_left (by norm_num), log_le_log_iff (by positivity) (by positivity), div_le_iff (by positivity), <- mul_inv_le_iff (by positivity), <-ge_iff_le, mul_comm] at this\n    convert this using 2\n    convert zpow_neg K 17 using 1\n    norm_cast\n  calc (dimension B : \u211d)\n    _ \u2264 (((max (dimension A') (dimension A'')) : \u2115) : \u211d) := by norm_cast\n    _ \u2264 (40 / log 2) * d\u1d64[A # A] := hdim\n    _ \u2264 (40 / log 2) * log K := mul_le_mul_of_nonneg_left hdist (by positivity)", "proofType": "tactic", "proofLengthLines": 47, "proofLengthTokens": 2092}}
{"srcContext": "import Mathlib.Algebra.Order.Ring.Defs\nimport Mathlib.Algebra.Quotient\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.GroupTheory.Torsion\nimport Mathlib.LinearAlgebra.FreeModule.PID\nimport PFR.Mathlib.Data.Set.Pointwise.SMul\nimport PFR.EntropyPFR\nimport PFR.ImprovedPFR\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Mathlib.GroupTheory.Torsion\n\n/-!\n# Weak PFR over the integers\n\nHere we use the entropic form of PFR to deduce a weak form of PFR over the integers.\n\n## Main statement\n\n* `weak_PFR_int`: Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A+A\\rvert\\leq K\\lvert A\\rvert$.\n  There exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$ and\n  $\\dim A' \\leq (40/\\log 2)\\log K$.\n\n-/\n\nsection AddCommGroup\nvariable {G : Type*} [AddCommGroup G] {A B : Set G}\n\nopen Set\nopen scoped Pointwise\n\ndef IsShift (A B : Set G) : Prop := \u2203 x : G, A = x +\u1d65 B\n\nlemma IsShift.sub_self_congr : IsShift A B \u2192 A - A = B - B := by\n  rintro \u27e8x, rfl\u27e9; simp [vadd_sub_vadd_comm, singleton_zero]\n\nlemma IsShift.card_congr : IsShift A B \u2192 Nat.card A = Nat.card B := by rintro \u27e8x, rfl\u27e9; simp\n\n/-- The property of two sets A, B of a group G not being contained in cosets of the same proper\nsubgroup -/\ndef NotInCoset (A B : Set G) : Prop := AddSubgroup.closure ((A - A) \u222a (B - B)) = \u22a4\n\n/-- Without loss of generality, one can move (up to translation and embedding) any pair A, B of non-empty sets into a subgroup where they are not in a coset. -/\nlemma wlog_notInCoset (hA : A.Nonempty) (hB : B.Nonempty) :\n    \u2203 (G' : AddSubgroup G) (A' B' : Set G'), IsShift A A' \u2227 IsShift B B' \u2227 NotInCoset A' B' := by\n  obtain \u27e8x, hx\u27e9 := hA\n  obtain \u27e8y, hy\u27e9 := hB\n  set G' := AddSubgroup.closure ((A - A) \u222a (B - B))\n  set A' : Set G' := (\u2191) \u207b\u00b9' ((-x) +\u1d65 A)\n  set B' : Set G' := (\u2191) \u207b\u00b9' ((-y) +\u1d65 B)\n  have hxA : -x +\u1d65 A \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hx).trans $ (subset_union_left ..).trans\n      AddSubgroup.subset_closure\n  have hyB : -y +\u1d65 B \u2286 range ((\u2191) : G' \u2192 G) := by\n    simp only [\u2190 singleton_add', \u2190 neg_singleton, neg_add_eq_sub, SetLike.coe_sort_coe,\n      Subtype.range_coe_subtype, SetLike.mem_coe]\n    exact (sub_subset_sub_left $ singleton_subset_iff.2 hy).trans $ (subset_union_right ..).trans\n      AddSubgroup.subset_closure\n  have hA : IsShift A A' := \u27e8x, by rw [Set.image_preimage_eq_of_subset hxA, vadd_neg_vadd]\u27e9\n  have hB : IsShift B B' := \u27e8y, by rw [Set.image_preimage_eq_of_subset hyB, vadd_neg_vadd]\u27e9\n  refine \u27e8G', A', B', hA, hB, ?_\u27e9\n  unfold NotInCoset\n  convert AddSubgroup.closure_preimage_eq_top ((A - A) \u222a (B - B))\n  simp_rw [preimage_union, hA.sub_self_congr, hB.sub_self_congr]\n  rw [preimage_sub, preimage_sub]\n  simp [A', B', preimage_image_preimage]\n  all_goals apply_rules [Subtype.coe_injective, (image_preimage_subset ..).trans, hxA, hyB]\n\nend AddCommGroup\n\nsection Torsion\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [MeasurableSpace G] [MeasurableSingletonClass G]\n  [Countable G] {\u03a9 \u03a9' : Type u} [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  (\u03bc : Measure \u03a9 := by volume_tac) (\u03bc': Measure \u03a9' := by volume_tac)\n  [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\n/-- If $G$ is torsion-free and $X,Y$ are $G$-valued random variables then $d[X;2Y]\\leq 5d[X;Y]$.  -/\nlemma torsion_free_doubling [FiniteRange X] [FiniteRange Y]\n    (hX : Measurable X) (hY : Measurable Y) (hG : AddMonoid.IsTorsionFree G) :\n    d[X ; \u03bc # (Y + Y) ; \u03bc'] \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := by\n  obtain \u27e8A, mA, \u03bcA, X', Y'\u2081, Y'\u2082, h\u03bcA, h_indep, hX'_meas, hY'\u2081_meas, hY'\u2082_meas, hX'_ident,\n    hY'\u2081_ident, hY'\u2082_ident, _, _, _\u27e9 := independent_copies3_nondep_finiteRange hX hY hY \u03bc \u03bc' \u03bc'\n  have h_meas (i : Fin 3) : Measurable (![X', Y'\u2081, Y'\u2082] i) := by fin_cases i <;> assumption\n  haveI : NoZeroSMulDivisors \u2115 G := hG.noZeroNsmulDivisors\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] = H[\u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8x, y\u2081, y\u2082\u27e9 \u21a6 (x, y\u2081 - y\u2082, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', \u27e8Y'\u2081, Y'\u2082\u27e9\u27e9 ; \u03bcA] = _\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hX'_meas <| Measurable.prod hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun X' (prod Y'\u2081 Y'\u2082) \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 1 2 0 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hX'_meas (by exact Measurable.prod hY'\u2081_meas hY'\u2082_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2081_meas hY'\u2082_meas (h_indep.indepFun (show 1 \u2260 2 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, two_mul]\n  have : H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[X ; \u03bc] + H[Y ; \u03bc'] := calc\n    H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[\u27e8X', Y'\u2081\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u2081\u27e9 \u21a6 (x, x - 2 \u2022 y\u2081)\n      show H[f \u2218 \u27e8X', Y'\u2081\u27e9 ; \u03bcA] = _\n      apply entropy_comp_of_injective \u03bcA (by exact Measurable.prod hX'_meas hY'\u2081_meas) f\n      exact fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all [smul_right_inj]\n    _ = H[X ; \u03bc] + H[Y ; \u03bc'] := by\n      rw [IndepFun.entropy_pair_eq_add hX'_meas hY'\u2081_meas (h_indep.indepFun (show 0 \u2260 1 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq]\n  let f : G \u00d7 G \u2192 G \u00d7 G := fun \u27e8x, y\u27e9 \u21a6 (x, y - x)\n  have hf : f.Injective := fun \u27e8_, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _\u27e9 := h; simp_all\n  have : H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      show _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082 - (Y'\u2081 - Y'\u2082)\u27e9 ; \u03bcA]\n      rw [sub_sub_sub_cancel_right, \u2190 sub_add_eq_sub_sub, two_nsmul]\n    _ = H[\u27e8Y'\u2081 - Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n      refine entropy_comp_of_injective \u03bcA (Measurable.prod ?_ ?_) f hf\n      \u00b7 exact Measurable.sub hY'\u2081_meas hY'\u2082_meas\n      \u00b7 exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] :=\n      entropy_pair_le_add (hY'\u2081_meas.sub' hY'\u2082_meas) (hX'_meas.sub' hY'\u2081_meas |>.sub' hY'\u2082_meas) \u03bcA\n  have : H[\u27e8X', \u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9\u27e9 ; \u03bcA] + H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] \u2264\n      H[\u27e8X', X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] + H[\u27e8Y'\u2081 - Y'\u2082, X' - 2 \u2022 Y'\u2081\u27e9 ; \u03bcA] := by\n    haveI : FiniteRange (Y'\u2081 - Y'\u2082) := FiniteRange.sub Y'\u2081 Y'\u2082\n    haveI : FiniteRange (2 \u2022 Y'\u2081) := by show FiniteRange ((fun x \u21a6 2 \u2022 x) \u2218 Y'\u2081); infer_instance\n    apply entropy_triple_add_entropy_le \u03bcA hX'_meas (Measurable.sub hY'\u2081_meas hY'\u2082_meas)\n    exact Measurable.sub' hX'_meas <| Measurable.const_smul hY'\u2081_meas 2\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := calc\n    H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] := by\n      let f : G \u00d7 G \u00d7 G \u2192 G \u00d7 G \u00d7 G := fun \u27e8y\u2081, y\u2082, x\u27e9 \u21a6 (y\u2081, y\u2082, x - y\u2081 - y\u2082)\n      show H[f \u2218 \u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA] = H[\u27e8Y'\u2081, \u27e8Y'\u2082, X'\u27e9\u27e9 ; \u03bcA]\n      refine entropy_comp_of_injective \u03bcA ?_ f ?_\n      \u00b7 exact Measurable.prod hY'\u2081_meas <| Measurable.prod hY'\u2082_meas hX'_meas\n      \u00b7 exact fun \u27e8_, _, _\u27e9 _ h \u21a6 by simp [f] at h; obtain \u27e8_, _, _\u27e9 := h; simp_all\n    _ = H[X ; \u03bc] + 2 * H[Y ; \u03bc'] := by\n      have : IndepFun Y'\u2081 (prod Y'\u2082 X') \u03bcA := Indep.symm <|\n        h_indep.indepFun_prod_mk h_meas 2 0 1 (by decide) (by decide)\n      rw [this.entropy_pair_eq_add hY'\u2081_meas (by exact Measurable.prod hY'\u2082_meas hX'_meas),\n        IndepFun.entropy_pair_eq_add hY'\u2082_meas hX'_meas (h_indep.indepFun (show 2 \u2260 0 by decide)),\n        hX'_ident.entropy_eq, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq]\n      group\n  have : H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := calc\n    H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := by rw [sub_right_comm] ; rfl\n    _ = H[\u27e8Y'\u2081, X' - Y'\u2082\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2081_meas <| Measurable.sub hX'_meas hY'\u2082_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2082 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2082) := FiniteRange.sub X' Y'\u2082\n      convert IndepFun.entropy_pair_eq_add hY'\u2081_meas (hX'_meas.sub hY'\u2082_meas)\n        <| h_indep.indepFun_sub_right h_meas 1 0 2 (by decide) (by decide)\n      exact hY'\u2081_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := calc\n    H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] = H[f \u2218 \u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := rfl\n    _ = H[\u27e8Y'\u2082, X' - Y'\u2081\u27e9 ; \u03bcA] := entropy_comp_of_injective \u03bcA\n      (by exact Measurable.prod hY'\u2082_meas <| Measurable.sub hX'_meas hY'\u2081_meas) f hf\n    _ = H[Y ; \u03bc'] + H[X' - Y'\u2081 ; \u03bcA] := by\n      haveI : FiniteRange (X' - Y'\u2081) := FiniteRange.sub X' Y'\u2081\n      convert IndepFun.entropy_pair_eq_add hY'\u2082_meas (hX'_meas.sub hY'\u2081_meas)\n        <| h_indep.indepFun_sub_right h_meas 2 0 1 (by decide) (by decide)\n      exact hY'\u2082_ident.entropy_eq.symm\n  have : H[\u27e8Y'\u2081, \u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9\u27e9 ; \u03bcA] + H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264\n      H[\u27e8Y'\u2081, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] + H[\u27e8Y'\u2082, X' - Y'\u2081 - Y'\u2082\u27e9 ; \u03bcA] := by\n    apply entropy_triple_add_entropy_le \u03bcA hY'\u2081_meas hY'\u2082_meas\n    exact Measurable.sub (Measurable.sub hX'_meas hY'\u2081_meas) hY'\u2082_meas\n  have : H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := calc\n    H[X' - Y'\u2081 - Y'\u2082 ; \u03bcA] \u2264 H[X' - Y'\u2081 ; \u03bcA] + H[X' - Y'\u2082 ; \u03bcA] - H[X ; \u03bc] := by linarith\n    _ = 2 * d[X ; \u03bc # Y ; \u03bc'] + H[Y ; \u03bc'] := by\n      nth_rw 1 [two_mul, \u2190 hX'_ident.rdist_eq hY'\u2081_ident, \u2190 hX'_ident.rdist_eq hY'\u2082_ident]\n      have h1 : d[X' ; \u03bcA # Y'\u2081 ; \u03bcA] = H[X' - Y'\u2081 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2081 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 1 by decide)).rdist_eq hX'_meas hY'\u2081_meas\n      have h2 : d[X' ; \u03bcA # Y'\u2082 ; \u03bcA] = H[X' - Y'\u2082 ; \u03bcA] - H[X' ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 0 \u2260 2 by decide)).rdist_eq hX'_meas hY'\u2082_meas\n      rw [h1, h2, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, hX'_ident.entropy_eq]\n      group\n  have : d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] \u2264\n      d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := calc\n    d[X ; \u03bc # 2 \u2022 Y ; \u03bc'] = H[X' - 2 \u2022 Y'\u2081 ; \u03bcA] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by\n      have h2Y_ident : IdentDistrib (2 \u2022 Y'\u2081) (2 \u2022 Y) (\u03bc := \u03bcA) (\u03bd := \u03bc') := by\n        convert hY'\u2081_ident.comp <| measurable_discrete <| fun g \u21a6 2 \u2022 g\n      have h2Y_indep : IndepFun X' (2 \u2022 Y'\u2081) (\u03bc := \u03bcA) := by\n        convert (h_indep.indepFun (show 0 \u2260 1 by decide)).comp measurable_id\n          (measurable_const_smul 2)\n      rw [\u2190 hX'_ident.rdist_eq h2Y_ident,\n        h2Y_indep.rdist_eq hX'_meas <| Measurable.const_smul hY'\u2081_meas 2,\n        hX'_ident.entropy_eq, h2Y_ident.entropy_eq]\n    _ \u2264 H[Y'\u2081 - Y'\u2082 ; \u03bcA] + 2 * d[X ; \u03bc # Y ; \u03bc'] - H[X ; \u03bc] / 2 - H[2 \u2022 Y ; \u03bc'] / 2 := by linarith\n    _ = d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] + (H[Y ; \u03bc'] - H[X ; \u03bc]) / 2 + 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n      have H2Y : H[2 \u2022 Y ; \u03bc'] = H[Y ; \u03bc'] := by\n        let f (g : G) := 2 \u2022 g\n        exact entropy_comp_of_injective \u03bc' hY f (fun _ _ \u21a6 by simp [f, smul_right_inj])\n      have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] = H[Y'\u2081 - Y'\u2082 ; \u03bcA] - H[Y'\u2081 ; \u03bcA] / 2 - H[Y'\u2082 ; \u03bcA] / 2 :=\n        (h_indep.indepFun (show 1 \u2260 2 by decide)).rdist_eq hY'\u2081_meas hY'\u2082_meas\n      rw [this, hY'\u2081_ident.entropy_eq, hY'\u2082_ident.entropy_eq, H2Y]\n      group\n  have : d[Y'\u2081 ; \u03bcA # Y'\u2082 ; \u03bcA] \u2264 2 * d[X ; \u03bc # Y ; \u03bc'] := by\n    rw [two_mul]\n    convert rdist_triangle hY'\u2081_meas hX'_meas hY'\u2082_meas (\u03bc := \u03bcA) (\u03bc' := \u03bcA) (\u03bc'' := \u03bcA)\n    \u00b7 exact rdist_symm.trans (hY'\u2081_ident.rdist_eq hX'_ident).symm\n    \u00b7 exact (hX'_ident.rdist_eq hY'\u2082_ident).symm\n  rw [\u2190 two_nsmul]\n  linarith [abs_le.mp <| diff_ent_le_rdist hX hY (\u03bc := \u03bc) (\u03bc' := \u03bc')]\n\n/-- If $G$ is a torsion-free group and $X,Y$ are $G$-valued random variables and\n$\\phi : G\\to \\mathbb{F}_2^d$ is a homomorphism then\n\\[\\mathbb{H}(\\phi(X))\\leq 10d[X;Y].\\] -/\nlemma torsion_dist_shrinking {H : Type u} [FiniteRange X] [FiniteRange Y] (hX : Measurable X)\n  (hY : Measurable Y) [AddCommGroup H] [ElementaryAddCommGroup H 2]\n  [MeasurableSpace H] [MeasurableSingletonClass H] [Countable H]\n  (hG : AddMonoid.IsTorsionFree G) (\u03c6 : G \u2192+ H) :\n  H[\u03c6 \u2218 X ; \u03bc] \u2264 10 * d[X; \u03bc # Y ; \u03bc'] := by\n  have :=\n    calc d[\u03c6 \u2218 X ; \u03bc # \u03c6 \u2218 (Y + Y); \u03bc'] \u2264 d[X; \u03bc # (Y + Y) ; \u03bc'] := rdist_of_hom_le \u03c6 hX (Measurable.add hY hY)\n    _ \u2264 5 * d[X; \u03bc # Y ; \u03bc'] := torsion_free_doubling X Y \u03bc \u03bc' hX hY hG\n  have eq_zero : \u03c6 \u2218 (Y + Y) = fun _ \u21a6 0 := by ext x ; simp only [Function.comp_apply, Pi.add_apply,\n    map_add, ElementaryAddCommGroup.add_self]\n  rwa [eq_zero, rdist_zero_eq_half_ent, div_le_iff zero_lt_two, mul_assoc, mul_comm _ 2, \u2190mul_assoc,\n    show (5*2 : \u211d) = 10 by norm_num] at this\n\nend Torsion\n\ninstance {G : Type u} [AddCommGroup G] [Fintype G] [MeasurableSpace G] [MeasurableSingletonClass G] (H : AddSubgroup G)\n    : MeasurableSingletonClass (G \u29f8 H) :=\n  \u27e8\u03bb _ \u21a6 by { rw [measurableSet_quotient]; simp [measurableSet_discrete] }\u27e9\n\nsection F2_projection\n\nopen Real ProbabilityTheory MeasureTheory\n\nvariable {G : Type u} [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n[MeasurableSingletonClass G] {\u03a9 \u03a9' : Type*}\n\n/-- Let $G=\\mathbb{F}_2^n$ and $X,Y$ be $G$-valued random variables such that\n\\[\\mathbb{H}(X)+\\mathbb{H}(Y)> (20/\\alpha) d[X;Y],\\]\nfor some $\\alpha > 0$.\nThere is a non-trivial subgroup $H\\leq G$ such that\n\\[\\log \\lvert H\\rvert <(1+\\alpha)/2 (\\mathbb{H}(X)+\\mathbb{H}(Y))\\] and\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))< \\alpha (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nwhere $\\psi:G\\to G/H$ is the natural projection homomorphism.\n-/\nlemma app_ent_PFR' [MeasureSpace \u03a9] [MeasureSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n  {\u03b1 : \u211d} (hent : 20 * d[X # Y] < \u03b1 * (H[X] + H[Y])) (hX : Measurable X) (hY : Measurable Y) :\n  \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X] + H[Y]) \u2227\n  H[(QuotientAddGroup.mk' H) \u2218 X] + H[(QuotientAddGroup.mk' H) \u2218 Y] < \u03b1 * (H[X] + H[Y]) := by\n  let p : refPackage \u03a9 \u03a9' G := {\n    X\u2080\u2081 := X\n    X\u2080\u2082 := Y\n    hmeas1 := hX\n    hmeas2 := hY\n    \u03b7 := 1/8\n    h\u03b7 := by norm_num\n    h\u03b7' := by norm_num }\n  obtain \u27e8H, \u03a9'', h\u03a9'', U, _, hUmeas, hUunif, ineq\u27e9 := entropic_PFR_conjecture_improv p rfl\n  let \u03c8 := (QuotientAddGroup.mk' H)\n  use H\n  haveI : Finite H := Subtype.finite\n  -- Note that  H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y]\n  have ent_le : H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 20 * d[X # Y] := calc\n    H[\u03c8 \u2218 X] + H[\u03c8 \u2218 Y] \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 exact ent_of_proj_le hX hUmeas hUunif\n      \u00b7 exact ent_of_proj_le hY hUmeas hUunif\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- Note that (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y]\n  have log_sub_le : (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) \u2264 20 * d[X # Y] := calc\n    (log (Nat.card H) - H[X]) + (log (Nat.card H) - H[Y]) =\n      (H[U] - H[X]) + (H[U] - H[Y]) := by\n        rw [IsUniform.entropy_eq' hUunif hUmeas, SetLike.coe_sort_coe]\n    _ \u2264 |(H[U] - H[X])| + |(H[U] - H[Y])| := by gcongr <;> exact le_abs_self _\n    _ \u2264 2 * d[X # U] + 2 * d[Y # U] := by\n      gcongr\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hX\n      \u00b7 rw [rdist_symm]; exact diff_ent_le_rdist hUmeas hY\n    _ = 2 * (d[X # U] + d[Y # U]) := by ring\n    _ \u2264 2 * (10 * d[X # Y]) := by gcongr\n    _ = 20 * d[X # Y] := by ring\n  -- then the conclusion follows from the assumption `hent` and basic inequality manipulations\n  exact \u27e8by linarith, by linarith\u27e9\n\nvariable [MeasurableSpace \u03a9] [MeasurableSpace \u03a9'] (X : \u03a9 \u2192 G) (Y : \u03a9' \u2192 G)\n(\u03bc : Measure \u03a9 := by volume_tac) (\u03bc' : Measure \u03a9' := by volume_tac)\n[IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc']\n\nlemma app_ent_PFR (\u03b1 : \u211d) (hent: 20 * d[X;\u03bc # Y;\u03bc'] < \u03b1 * (H[X; \u03bc] + H[Y; \u03bc'])) (hX : Measurable X)\n    (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) < (1 + \u03b1) / 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc']\n      < \u03b1 * (H[ X; \u03bc] + H[Y; \u03bc']) :=\n  @app_ent_PFR' _ _ _ _ _ _ _ _ (MeasureSpace.mk \u03bc) (MeasureSpace.mk \u03bc') _ _ _ _ \u03b1 hent hX hY\n\nset_option maxHeartbeats 300000 in\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables and $\\alpha < 1$ then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq (1 + \u03b1) / (2 * (1 - \u03b1)) * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 20/\\alpha * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection'\n    (\u03b1 : \u211d) (hX : Measurable X) (hY : Measurable Y) (\u03b1pos : 0 < \u03b1) (\u03b1one : \u03b1 < 1) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X ; \u03bc] + H[Y ; \u03bc']) \u2227\n    \u03b1 * (H[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc']) \u2264\n      20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] := by\n  let S := { H : AddSubgroup G | (\u2203 (c : \u211d), 0 \u2264 c \u2227\n      log (Nat.card H) \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      c * (H[X; \u03bc] + H[Y;\u03bc'])) \u2227\n    20 * d[(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # (QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] <\n      \u03b1 * (H[ (QuotientAddGroup.mk' H) \u2218 X; \u03bc ] + H[ (QuotientAddGroup.mk' H) \u2218 Y; \u03bc']) }\n  have : 0 \u2264 H[X ; \u03bc] + H[Y ; \u03bc'] := by linarith [entropy_nonneg X \u03bc, entropy_nonneg Y \u03bc']\n  have : 0 < 1 - \u03b1 := sub_pos.mpr \u03b1one\n  by_cases hE : (\u22a5 : AddSubgroup G) \u2208 S\n  \u00b7 classical\n    obtain \u27e8H, \u27e8\u27e8c, hc, hlog, hup\u27e9, hent\u27e9, hMaxl\u27e9 :=\n      S.toFinite.exists_maximal_wrt id S (Set.nonempty_of_mem hE)\n    set \u03c8 : G \u2192+ G \u29f8 H := QuotientAddGroup.mk' H\n    have surj : Function.Surjective \u03c8 := QuotientAddGroup.mk'_surjective H\n\n    set G' := G \u29f8 H\n    have : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.quotient_group (by decide)\n      (by simp [AddSubgroup.zero_mem])\n\n    obtain \u27e8H', hlog', hup'\u27e9 := app_ent_PFR _ _ _ _ \u03b1 hent ((measurable_discrete _).comp hX)\n      ((measurable_discrete _).comp hY)\n    have H_ne_bot: H' \u2260 \u22a5 := by\n      by_contra!\n      rcases this with rfl\n      have inj : Function.Injective (QuotientAddGroup.mk' (\u22a5 : AddSubgroup G')) :=\n          (QuotientAddGroup.quotientBot : (G' \u29f8 \u22a5) \u2243+ G').symm.injective\n      rw [entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ inj,\n          entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ inj] at hup'\n      nlinarith [entropy_nonneg (\u03c8 \u2218 X) \u03bc, entropy_nonneg (\u03c8 \u2218 Y) \u03bc']\n    let H'' := H'.comap \u03c8\n    use H''\n\n    rw [\u2190 (AddSubgroup.map_comap_eq_self_of_surjective surj _ : H''.map \u03c8 = H')] at hup' hlog'\n    set H' := H''.map \u03c8\n\n    have Hlt : H < H'' := by\n      have : H = (\u22a5 : AddSubgroup G').comap \u03c8 := by\n        simp only [\u03c8, AddMonoidHom.comap_bot, QuotientAddGroup.ker_mk']\n      rw [this, AddSubgroup.comap_lt_comap_of_surjective surj]\n      exact Ne.bot_lt H_ne_bot\n\n    let \u03c6 : G' \u29f8 H' \u2243+ G \u29f8 H'' := QuotientAddGroup.quotientQuotientEquivQuotient H H'' Hlt.le\n    set \u03c8' : G' \u2192+ G' \u29f8 H' := QuotientAddGroup.mk' H'\n    set \u03c8'' : G \u2192+ G \u29f8 H'' := QuotientAddGroup.mk' H''\n    have diag : \u03c8' \u2218 \u03c8 = \u03c6.symm \u2218 \u03c8'' := rfl\n    rw [\u2190 Function.comp.assoc, \u2190 Function.comp.assoc, diag, Function.comp.assoc,\n        Function.comp.assoc] at hup'\n\n    have cond : log (Nat.card H'') \u2264\n        (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := by\n      have cardprod : Nat.card H'' = Nat.card H' * Nat.card H := by\n        have hcard\u2080 := Nat.card_congr <| (AddSubgroup.addSubgroupOfEquivOfLe Hlt.le).toEquiv\n        have hcard\u2081 := Nat.card_congr <|\n          (QuotientAddGroup.quotientKerEquivRange (\u03c8.restrict H'')).toEquiv\n        have hcard\u2082 := AddSubgroup.card_eq_card_quotient_mul_card_addSubgroup (H.addSubgroupOf H'')\n        rw [\u03c8.ker_restrict H'', QuotientAddGroup.ker_mk', \u03c8.restrict_range H''] at hcard\u2081\n        simpa only [\u2190 Nat.card_eq_fintype_card, hcard\u2080, hcard\u2081] using hcard\u2082\n      calc\n          log (Nat.card H'')\n      _ = log ((Nat.card H' : \u211d) * (Nat.card H : \u211d)) := by rw [cardprod]; norm_cast\n      _ = log (Nat.card H') + log (Nat.card H) := by\n        rw [Real.log_mul (Nat.cast_ne_zero.2 (@Nat.card_pos H').ne')\n              (Nat.cast_ne_zero.2 (@Nat.card_pos H).ne')]\n      _ \u2264 (1 + \u03b1) / 2 * (H[\u21d1\u03c8 \u2218 X ; \u03bc] + H[\u21d1\u03c8 \u2218 Y ; \u03bc']) + log (Nat.card H) := by gcongr\n      _ \u2264 (1 + \u03b1) / 2 * (c * (H[X; \u03bc] + H[Y;\u03bc'])) +\n            (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by gcongr\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by\n        field_simp; ring\n\n    have HS : H'' \u2209 S := \u03bb Hs => Hlt.ne (hMaxl H'' Hs Hlt.le)\n    simp only [S, Set.mem_setOf_eq, not_and, not_lt] at HS\n    refine \u27e8?_, HS \u27e8\u03b1 * c, by positivity, cond, ?_\u27e9\u27e9\n    \u00b7 calc\n      log (Nat.card H'')\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * (1 - \u03b1 * c) * (H[X; \u03bc] + H[Y;\u03bc']) := cond\n      _ \u2264 (1 + \u03b1) / (2 * (1 - \u03b1)) * 1 * (H[X; \u03bc] + H[Y;\u03bc']) := by gcongr; simp; positivity\n      _ = (1 + \u03b1) / (2 * (1 - \u03b1)) * (H[X; \u03bc] + H[Y;\u03bc']) := by simp only [mul_one]\n    \u00b7 calc\n      H[ \u03c8'' \u2218 X; \u03bc ] + H[ \u03c8'' \u2218 Y; \u03bc' ]\n      _ = H[ \u03c6.symm \u2218 \u03c8'' \u2218 X; \u03bc ] + H[ \u03c6.symm \u2218 \u03c8'' \u2218 Y; \u03bc' ] := by\n        simp_rw [\u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hX) _ \u03c6.symm.injective,\n                 \u2190 entropy_comp_of_injective _ ((measurable_discrete _).comp hY) _ \u03c6.symm.injective]\n      _ \u2264 \u03b1 * (H[ \u03c8 \u2218 X; \u03bc ] + H[ \u03c8 \u2218 Y; \u03bc' ]) := hup'.le\n      _ \u2264 \u03b1 * (c * (H[X ; \u03bc] + H[Y ; \u03bc'])) := by gcongr\n      _ = (\u03b1 * c) * (H[X ; \u03bc] + H[Y ; \u03bc']) := by ring\n  \u00b7 use \u22a5\n    constructor\n    \u00b7 simp only [AddSubgroup.mem_bot, Nat.card_eq_fintype_card, Fintype.card_ofSubsingleton,\n        Nat.cast_one, log_one]\n      positivity\n    \u00b7 simp only [S, Set.mem_setOf_eq, not_and, not_lt] at hE\n      exact hE \u27e81, by norm_num, by\n        norm_num; exact add_le_add (entropy_comp_le \u03bc hX _) (entropy_comp_le \u03bc' hY _)\u27e9\n\n/-- If $G=\\mathbb{F}_2^d$ and $X,Y$ are $G$-valued random variables then there is\na subgroup  $H\\leq \\mathbb{F}_2^d$ such that\n\\[\\log \\lvert H\\rvert \\leq 2 * (\\mathbb{H}(X)+\\mathbb{H}(Y))\\]\nand if $\\psi:G \\to G/H$ is the natural projection then\n\\[\\mathbb{H}(\\psi(X))+\\mathbb{H}(\\psi(Y))\\leq 34 * d[\\psi(X);\\psi(Y)].\\] -/\nlemma PFR_projection (hX : Measurable X) (hY : Measurable Y) :\n    \u2203 H : AddSubgroup G, log (Nat.card H) \u2264 2 * (H[X; \u03bc] + H[Y;\u03bc']) \u2227\n    H[(QuotientAddGroup.mk' H) \u2218 X; \u03bc] + H[(QuotientAddGroup.mk' H) \u2218 Y; \u03bc'] \u2264\n      34 * d[(QuotientAddGroup.mk' H) \u2218 X;\u03bc # (QuotientAddGroup.mk' H) \u2218 Y;\u03bc'] := by\n  rcases PFR_projection' X Y \u03bc \u03bc' ((3 : \u211d) / 5) hX hY (by norm_num) (by norm_num) with \u27e8H, h, h'\u27e9\n  refine \u27e8H, ?_, ?_\u27e9\n  \u00b7 convert h\n    norm_num\n  \u00b7 have : 0 \u2264 d[\u21d1(QuotientAddGroup.mk' H) \u2218 X ; \u03bc # \u21d1(QuotientAddGroup.mk' H) \u2218 Y ; \u03bc'] :=\n      rdist_nonneg ((measurable_discrete _).comp hX) ((measurable_discrete _).comp hY)\n    linarith\n\nend F2_projection\n\nopen MeasureTheory ProbabilityTheory Real Set\nopen scoped BigOperators\n\nlemma four_logs {a b c d : \u211d} (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) :\n    log ((a*b)/(c*d)) = log a + log b - log c - log d := by\n  rw [log_div, log_mul, log_mul, sub_sub] <;> positivity\n\nlemma sum_prob_preimage {G H : Type*} {X : Finset H} {A : Set G} [Finite A] {\u03c6 : A \u2192 X}\n    {A_ : H \u2192 Set G} (hA : A.Nonempty) (h\u03c6 : \u2200 x : X, A_ x = Subtype.val '' (\u03c6 \u207b\u00b9' {x})) :\n    \u2211 x in X, (Nat.card (A_ x) : \u211d) / (Nat.card A) = 1 := by\n  rw [\u2190 Finset.sum_div]\n  apply (div_eq_one_iff_eq <| Nat.cast_ne_zero.mpr <| Nat.pos_iff_ne_zero.mp (@Nat.card_pos _ hA.to_subtype _)).mpr\n  classical\n  haveI := Fintype.ofFinite A\n  rewrite [Nat.card_eq_fintype_card, \u2190 Finset.card_univ, Finset.card_eq_sum_card_fiberwise\n    <| fun a _ \u21a6 Finset.mem_univ (\u03c6 a), \u2190 Finset.sum_coe_sort]\n  norm_cast\n  congr; ext\n  rewrite [\u2190 Set.Finite.toFinset_setOf, (Set.toFinite _).card_toFinset, \u2190 Nat.card_eq_fintype_card,\n    h\u03c6, Nat.card_image_of_injective Subtype.val_injective]; rfl\n\n/-- Let $\\phi : G\\to H$ be a homomorphism and $A,B\\subseteq G$ be finite subsets. If $x,y\\in H$ then let $A_x=A\\cap \\phi^{-1}(x)$ and $B_y=B\\cap \\phi^{-1}(y)$. There exist $x,y\\in H$ such that $A_x,B_y$ are both non-empty and\n\\[d[\\phi(U_A);\\phi(U_B)]\\log \\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A_x\\rvert\\lvert B_y\\rvert}\\leq (\\mathbb{H}(\\phi(U_A))+\\mathbb{H}(\\phi(U_B)))(d(U_A,U_B)-d(U_{A_x},U_{B_y}).\\] -/\nlemma single_fibres {G H \u03a9 \u03a9': Type u}\n    [AddCommGroup G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n    [AddCommGroup H] [Countable H] [MeasurableSpace H] [MeasurableSingletonClass H]\n    [MeasureSpace \u03a9] [MeasureSpace \u03a9']\n    [IsProbabilityMeasure (\u2119 : Measure \u03a9)] [IsProbabilityMeasure (\u2119 : Measure \u03a9')]\n    (\u03c6 : G \u2192+ H)\n    {A B : Set G} [Finite A] [Finite B] {UA : \u03a9 \u2192 G} {UB: \u03a9' \u2192 G} (hA : A.Nonempty) (hB : B.Nonempty)\n    (hUA': Measurable UA) (hUB': Measurable UB) (hUA: IsUniform A UA) (hUB: IsUniform B UB)\n    (hUA_mem : \u2200 \u03c9, UA \u03c9 \u2208 A) (hUB_mem : \u2200 \u03c9, UB \u03c9 \u2208 B) :\n    \u2203 (x y : H) (Ax By: Set G),\n    Ax = A \u2229 \u03c6.toFun \u207b\u00b9' {x} \u2227 By = B \u2229 \u03c6.toFun \u207b\u00b9' {y} \u2227 Ax.Nonempty \u2227 By.Nonempty \u2227\n    d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB]\n    * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) \u2264\n    (H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n  have : Nonempty A := hA.to_subtype\n  have : Nonempty B := hB.to_subtype\n  haveI : FiniteRange UA := finiteRange_of_finset UA A.toFinite.toFinset (by simpa)\n  haveI : FiniteRange UB := finiteRange_of_finset UB B.toFinite.toFinset (by simpa)\n  have hUA_coe : IsUniform A.toFinite.toFinset.toSet UA := by rwa [Set.Finite.coe_toFinset]\n  have hUB_coe : IsUniform B.toFinite.toFinset.toSet UB := by rwa [Set.Finite.coe_toFinset]\n\n  let A_ (x : H) : Set G := A \u2229 \u03c6.toFun \u207b\u00b9' {x}\n  let B_ (y : H) : Set G := B \u2229 \u03c6.toFun \u207b\u00b9' {y}\n  let X : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UA)\n  let Y : Finset H := FiniteRange.toFinset (\u03c6.toFun \u2218 UB)\n\n  haveI h_Ax (x : X) : Nonempty (A_ x.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp x.property\n    use UA \u03c9; exact Set.mem_inter (hUA_mem \u03c9) (by exact h\u03c9)\n  haveI h_By (y : Y) : Nonempty (B_ y.val) := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := (FiniteRange.mem_iff _ _).mp y.property\n    use UB \u03c9; exact Set.mem_inter (hUB_mem \u03c9) (by exact h\u03c9)\n  have h_AX (a : A) : \u03c6.toFun a.val \u2208 X := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUA_coe.nonempty_preimage_of_mem hUA' (A.toFinite.mem_toFinset.mpr a.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun a.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n  have h_BY (b : B) : \u03c6.toFun b.val \u2208 Y := by\n    obtain \u27e8\u03c9, h\u03c9\u27e9 := hUB_coe.nonempty_preimage_of_mem hUB' (B.toFinite.mem_toFinset.mpr b.property)\n    exact (FiniteRange.mem_iff _ (\u03c6.toFun b.val)).mpr \u27e8\u03c9, congr_arg _ h\u03c9\u27e9\n\n  let \u03c6_AX (a : A) : X := by use \u03c6.toFun a.val; exact h_AX a\n  let \u03c6_BY (b : B) : Y := by use \u03c6.toFun b.val; exact h_BY b\n  have h_\u03c6_AX (x : X) : A_ x.val = \u03c6_AX \u207b\u00b9' {x} := by ext; simp [A_, \u03c6_AX]; simp [Subtype.ext_iff]\n  have h_\u03c6_BY (y : Y) : B_ y.val = \u03c6_BY \u207b\u00b9' {y} := by ext; simp [B_, \u03c6_BY]; simp [Subtype.ext_iff]\n\n  let p (x : H) (y : H) : \u211d :=\n    (Nat.card (A_ x).Elem) * (Nat.card (B_ y).Elem) / ((Nat.card A.Elem) * (Nat.card B.Elem))\n  have :\n    \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] :=\n  calc\n    _ = d[UA | \u03c6.toFun \u2218 UA # UB | \u03c6.toFun \u2218 UB] := by\n      rewrite [condRuzsaDist_eq_sum hUA' ((measurable_discrete _).comp hUA')\n        hUB' ((measurable_discrete _).comp hUB')]\n      refine Finset.sum_congr rfl <| fun x hx \u21a6 Finset.sum_congr rfl <| fun y hy \u21a6 ?_\n      haveI : Nonempty (A_ x) := h_Ax \u27e8x, hx\u27e9\n      haveI : Nonempty (B_ y) := h_By \u27e8y, hy\u27e9\n      let \u03bcx := (\u2119 : Measure \u03a9)[|(\u03c6.toFun \u2218 UA) \u207b\u00b9' {x}]\n      let \u03bcy := (\u2119 : Measure \u03a9')[|(\u03c6.toFun \u2218 UB) \u207b\u00b9' {y}]\n      have h_\u03bc_p : IsProbabilityMeasure \u03bcx \u2227 IsProbabilityMeasure \u03bcy := by\n        constructor <;> apply ProbabilityTheory.cond_isProbabilityMeasure <;> rw [Set.preimage_comp]\n        refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUA_coe hUA' _ ?_\n        swap; refine @IsUniform.measure_preimage_ne_zero _ _ _ _ _ _ _ _ _ _ hUB_coe hUB' _ ?_\n        all_goals rwa [Set.inter_comm, Set.Finite.coe_toFinset]\n      have h_\u03bc_unif : IsUniform (A_ x) UA \u03bcx \u2227 IsUniform (B_ y) UB \u03bcy := by\n        have : _ \u2227 _ := \u27e8hUA.restrict hUA' (\u03c6.toFun \u207b\u00b9' {x}), hUB.restrict hUB' (\u03c6.toFun \u207b\u00b9' {y})\u27e9\n        rwa [Set.inter_comm _ A, Set.inter_comm _ B] at this\n      rewrite [rdist_set_eq_rdist h_\u03bc_p.1 h_\u03bc_p.2 h_\u03bc_unif.1 h_\u03bc_unif.2 hUA' hUB']\n      show _ = (Measure.real _ (UA \u207b\u00b9' (_ \u207b\u00b9' _))) * (Measure.real _ (UB \u207b\u00b9' (_ \u207b\u00b9' _))) * _\n      rewrite [hUA_coe.measureReal_preimage hUA', hUB_coe.measureReal_preimage hUB']\n      simp_rw [p, A_, B_, IsProbabilityMeasure.measureReal_univ, one_mul]\n      rewrite [mul_div_mul_comm, Set.inter_comm A, Set.inter_comm B]\n      simp only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, Finset.mem_val]; rfl\n    _ \u2264 d[UA # UB] - d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      rewrite [ZeroHom.toFun_eq_coe, AddMonoidHom.toZeroHom_coe]\n      linarith only [rdist_le_sum_fibre \u03c6 hUA' hUB' (\u03bc := \u2119) (\u03bc' := \u2119)]\n  let M := H[\u03c6.toFun \u2218 UA] + H[\u03c6.toFun \u2218 UB]\n  have hM : M = \u2211 x in X, \u2211 y in Y, Real.negMulLog (p x y) := by\n    have h_compl {x y} (h_notin : (x, y) \u2209 X \u00d7\u02e2 Y) : Real.negMulLog (p x y) = 0 := by\n      unfold_let p; beta_reduce\n      rewrite [Finset.mem_product, not_and_or] at h_notin\n      suffices A_ x = \u2205 \u2228 B_ y = \u2205 by obtain h | h := this <;> rw [h] <;> simp\n      refine h_notin.imp ?_ ?_\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_AX \u27e8a, ha\u27e9)\n      \u00b7 rw [\u2190 not_nonempty_iff_eq_empty]\n        rintro h \u27e8a, ha, rfl\u27e9\n        exact h (h_BY \u27e8a, ha\u27e9)\n    unfold_let M\n    unfold entropy\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUA').aemeasurable\n    haveI := isProbabilityMeasure_map (\u03bc := \u2119) ((measurable_discrete \u03c6).comp hUB').aemeasurable\n    rewrite [\u2190 Finset.sum_product', \u2190 tsum_eq_sum fun _ \u21a6 h_compl, \u2190 measureEntropy_prod]\n    apply tsum_congr; intro; congr\n    rewrite [\u2190 Set.singleton_prod_singleton, Measure.smul_apply, Measure.prod_prod,\n      Measure.map_apply ((measurable_discrete _).comp hUA') (MeasurableSet.singleton _),\n      Measure.map_apply ((measurable_discrete _).comp hUB') (MeasurableSet.singleton _),\n      Set.preimage_comp, hUA_coe.measure_preimage hUA',\n      Set.preimage_comp, hUB_coe.measure_preimage hUB']\n    simp [p, A_, B_, mul_div_mul_comm, Set.inter_comm, ENNReal.toReal_div]\n  have h_sum : \u2211 x in X, \u2211 y in Y,\n      (p x y) * (M * d\u1d64[A_ x # B_ y] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x y)) \u2264\n      M * d[UA # UB] :=\n  calc\n    _ = \u2211 x in X, \u2211 y in Y, (p x y) * M * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [hM, Finset.sum_mul, \u2190 Finset.sum_add_distrib]\n      refine Finset.sum_congr rfl <| fun _ _ \u21a6 Finset.sum_congr rfl <| fun _ _ \u21a6 ?_\n      simp only [negMulLog, left_distrib, mul_assoc, Finset.sum_mul]\n      exact congrArg (HAdd.hAdd _) (by group)\n    _ = M * \u2211 x in X, \u2211 y in Y, (p x y) * d\u1d64[A_ x # B_ y] + M * d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] := by\n      simp_rw [Finset.mul_sum]\n      congr; ext; congr; ext; group\n    _ \u2264 M * d[UA # UB] := by\n      rewrite [\u2190 left_distrib]\n      apply mul_le_mul_of_nonneg_left\n      \u00b7 linarith\n      \u00b7 unfold_let M\n        linarith only [entropy_nonneg (\u03c6.toFun \u2218 UA) \u2119, entropy_nonneg (\u03c6.toFun \u2218 UB) \u2119]\n  have : \u2203 x : X, \u2203 y : Y,\n      M * d\u1d64[A_ x.val # B_ y.val] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p x.val y.val) \u2264\n      M * d[UA # UB] := by\n    let f (xy : H \u00d7 H) := (p xy.1 xy.2) * (M * d[UA # UB])\n    let g (xy : H \u00d7 H) := (p xy.1 xy.2) *\n      (M * d\u1d64[A_ xy.1 # B_ xy.2] + d[\u03c6.toFun \u2218 UA # \u03c6.toFun \u2218 UB] * -Real.log (p xy.1 xy.2))\n    by_contra hc; push_neg at hc\n    replace hc : \u2200 xy \u2208 X \u00d7\u02e2 Y, f xy < g xy := by\n      refine fun xy h \u21a6 mul_lt_mul_of_pos_left ?_ ?_\n      \u00b7 exact hc \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9 \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n      \u00b7 haveI : Nonempty _ := h_Ax \u27e8xy.1, (Finset.mem_product.mp h).1\u27e9\n        haveI : Nonempty _ := h_By \u27e8xy.2, (Finset.mem_product.mp h).2\u27e9\n        simp only [p, div_pos, mul_pos, Nat.cast_pos, Nat.card_pos]\n    have h_nonempty : Finset.Nonempty (X \u00d7\u02e2 Y) := by\n      use \u27e8\u03c6.toFun <| UA <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9,\n        \u03c6.toFun <| UB <| Classical.choice <| ProbabilityMeasure.nonempty \u27e8\u2119, inferInstance\u27e9\u27e9\n      exact Finset.mem_product.mpr \u27e8FiniteRange.mem _ _, FiniteRange.mem _ _\u27e9\n    replace hc := Finset.sum_lt_sum_of_nonempty h_nonempty hc\n    have h_p_one : \u2211 x in X \u00d7\u02e2 Y, p x.1 x.2 = 1 := by\n      simp_rw [Finset.sum_product, p, mul_div_mul_comm, \u2190 Finset.mul_sum,\n        \u2190 sum_prob_preimage hA h_\u03c6_AX, sum_prob_preimage hB h_\u03c6_BY, mul_one]\n    rewrite [\u2190 Finset.sum_mul, h_p_one, one_mul, Finset.sum_product] at hc\n    exact not_le_of_gt hc h_sum\n  obtain \u27e8x, y, hxy\u27e9 := this\n  refine \u27e8x, y, A_ x.val, B_ y.val, rfl, rfl, @nonempty_of_nonempty_subtype _ _ (h_Ax x), @nonempty_of_nonempty_subtype _ _ (h_By y), ?_\u27e9\n  rewrite [\u2190 inv_div, Real.log_inv]\n  show _ * -log (p x.val y.val) \u2264 M * _\n  linarith only [hxy]\n\nsection dim\n\nopen Classical\n\nvariable {G : Type*} [AddCommGroup G]\n\n/- If G \u2245 \u2124\u1d48 then there is a subgroup H of G such that A lies in a coset of H. This is helpful to\n  give the equivalent definition of `dimension`. Here this is stated in greated generality since the\n  proof carries over automatically-/\nlemma exists_coset_cover (A : Set G) :\n    \u2203 (d : \u2115), \u2203 (S : Submodule \u2124 G) (v : G),\n      FiniteDimensional.finrank \u2124 S = d \u2227 \u2200 a \u2208 A, a - v \u2208 S := by\n  existsi FiniteDimensional.finrank \u2124 (\u22a4 : Submodule \u2124 G), \u22a4, 0\n  refine \u27e8rfl, fun a _ \u21a6 trivial\u27e9\n\n/-- The dimension of the affine span over `\u2124` of a subset of an additive group. -/\nnoncomputable def dimension (A : Set G) : \u2115 := Nat.find (exists_coset_cover A)\n\nlemma dimension_le_of_coset_cover (A : Set G) (S : Submodule \u2124 G) (v : G)\n    (hA : \u2200 a \u2208 A, a - v \u2208 S) : dimension A \u2264 FiniteDimensional.finrank \u2124 S := by\n  apply Nat.find_le\n  existsi S , v\n  exact \u27e8rfl, hA\u27e9\n\nlemma dimension_le_rank [Module.Finite \u2124 G] (A : Set G) :\n    dimension A \u2264 FiniteDimensional.finrank \u2124 G := by\n  obtain \u27e8S, v, hs, _\u27e9 := Nat.find_spec (exists_coset_cover A)\n  rw [dimension, \u2190hs]\n  apply Submodule.finrank_le S\n\nend dim\n\nvariable {G : Type u} [AddCommGroup G] [Module.Free \u2124 G] [Module.Finite \u2124 G] [Countable G] [MeasurableSpace G] [MeasurableSingletonClass G]\n\nopen Real MeasureTheory ProbabilityTheory Pointwise Set\n\n/-- Move to Mathlib? `Finsupp.mapRange` of a surjective function is surjective. -/\nlemma Finsupp.mapRange_surjective {\u03b1 : Type u_1} {M : Type u_5} {N : Type u_7} [Zero M] [Zero N] (f : M \u2192 N) (hf : f 0 = 0)\n  (hs : Function.Surjective f) : Function.Surjective (Finsupp.mapRange (\u03b1 := \u03b1) f hf) := by\n  classical\n  let g (n : N) : M := if n = 0 then 0 else Function.surjInv hs n\n  have : Function.RightInverse g f := by\n    intro n\n    by_cases h : n = 0\n    . simp [g, h, hf]\n    \u00b7 simp [g, h, Function.surjInv_eq hs n]\n  have hg : g 0 = 0 := by simp [g]\n  have hfg : (f \u2218 g) 0 = 0 := by simp [hf, hg]\n  intro F\n  use Finsupp.mapRange g hg F\n  rw [\u2190 Finsupp.mapRange_comp (h := hfg)]\n  convert Finsupp.mapRange_id F\n  convert Function.RightInverse.id this\n\n/-- A free Z-module is torsion-free. Move to Mathlib? -/\nlemma torsion_free : AddMonoid.IsTorsionFree G := by\n    rintro x hx hn\n    rw [isOfFinAddOrder_iff_nsmul_eq_zero] at hn\n    rcases hn with \u27e8 n, hn, hn' \u27e9\n    apply_fun Module.Free.repr \u2124 G at hn'\n    simp_rw [map_nsmul, map_zero, smul_eq_zero, AddEquivClass.map_eq_zero_iff, hx, or_false] at hn'\n    linarith\n\n/-- If G is a rank n free Z-module, then G/2G is a finite elementary 2-group of cardinality 2^n.  Code is slow, needs to be golfed -/\nlemma weak_PFR_quotient_prelim :\n  let H := G \u29f8 (AddMonoidHom.range (zsmulAddGroupHom 2))\n  ElementaryAddCommGroup H 2 \u2227 Finite H \u2227 Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  set \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  have hH_elem : ElementaryAddCommGroup H 2 := by\n    apply ElementaryAddCommGroup.quotient_group (by decide)\n    intro x; rw [AddMonoidHom.mem_range]\n    use x\n    rw [zsmulAddGroupHom_apply]\n    norm_cast\n  let B := Module.Free.ChooseBasisIndex \u2124 G\n  let bG : Basis B \u2124 G := Module.Free.chooseBasis \u2124 G\n  have hB_fin : Fintype B := by infer_instance\n  have hB_card : Nat.card B = FiniteDimensional.finrank \u2124 G := by\n    rw [FiniteDimensional.finrank_eq_card_basis bG, Nat.card_eq_fintype_card]\n  have hH_module : Module (ZMod 2) H := by infer_instance\n  let mod : (B \u2192\u2080 \u2124) \u2192+ (B \u2192\u2080 ZMod 2) := Finsupp.mapRange.addMonoidHom (Int.castAddHom (ZMod 2))\n  let f : G \u2192+ (B \u2192\u2080 \u2124) := bG.repr\n  have hker : G\u2082 \u2264 AddMonoidHom.ker (AddMonoidHom.comp mod f) := by\n    intro x hx\n    simp only [AddMonoidHom.mem_range, G\u2082, \u03c8, zsmulAddGroupHom_apply] at hx\n    simp_rw [AddMonoidHom.mem_ker, AddMonoidHom.coe_comp, Function.comp_apply, mod,\n      Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom]\n    rcases hx with \u27e8y, rfl\u27e9\n    ext b\n    simp_rw [map_zsmul, Finsupp.mapRange_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul,\n      Int.cast_mul, Int.cast_ofNat, Finsupp.coe_zero, Pi.zero_apply, mul_eq_zero]\n    left\n    exact ZMod.nat_cast_self 2\n  let g : H \u2192+ (B \u2192\u2080 ZMod 2) := QuotientAddGroup.lift G\u2082 (AddMonoidHom.comp mod f) hker\n  have hsur : Function.Surjective g := by\n    have h1 : Function.Surjective mod := Finsupp.mapRange_surjective (Int.castAddHom (ZMod 2)) (map_zero _) ZMod.int_cast_surjective\n    have h2 := h1.comp bG.repr.surjective\n    have h3 : mod \u2218 bG.repr = g \u2218 (QuotientAddGroup.mk' G\u2082) := by\n      ext x b\n      simp_rw [mod, Function.comp_apply, Finsupp.mapRange.addMonoidHom_apply, Int.coe_castAddHom,\n        Finsupp.mapRange_apply, QuotientAddGroup.coe_mk', g]\n      rw [QuotientAddGroup.lift_mk]\n      simp [mod, f]\n    rw [h3] at h2\n    apply Function.Surjective.of_comp h2\n  have hinj : Function.Injective g := by\n    rw [injective_iff_map_eq_zero]\n    intro x hx\n    rcases QuotientAddGroup.mk'_surjective G\u2082 x with \u27e8y, rfl\u27e9\n    simp only [QuotientAddGroup.mk'_apply, g] at hx\n    rw [QuotientAddGroup.lift_mk] at hx\n    simp_rw [AddMonoidHom.coe_comp, Function.comp_apply, mod, Finsupp.mapRange.addMonoidHom_apply,\n      Int.coe_castAddHom, DFunLike.ext_iff,Finsupp.mapRange_apply, Finsupp.coe_zero, Pi.zero_apply,\n      ZMod.int_cast_zmod_eq_zero_iff_dvd] at hx\n    replace hx := fun x \u21a6 Int.mul_ediv_cancel' (hx x)\n    let z (b : B) := ((Module.Free.chooseBasis \u2124 G).repr y) b / 2\n    let z' := (Finsupp.equivFunOnFinite).symm z\n    change \u2200 b : B, 2 * z' b = (f y) b at hx\n    let x' := bG.repr.symm z'\n    rw [QuotientAddGroup.mk'_apply, QuotientAddGroup.eq_zero_iff, AddMonoidHom.mem_range]\n    simp_rw [\u03c8, zsmulAddGroupHom_apply]\n    use x'\n    change 2 \u2022 (bG.repr.symm.toLinearMap.toAddMonoidHom z') = y\n    rw [\u2190 AddMonoidHom.map_zsmul bG.repr.symm.toLinearMap.toAddMonoidHom z' (2 : \u2124)]\n    rw [\u2190 LinearEquiv.symm_apply_apply bG.repr y]\n    change bG.repr.symm (2 \u2022 z') = bG.repr.symm (f y)\n    congr\n    ext b\n    rw [Finsupp.smul_apply, \u2190 hx b, smul_eq_mul]\n  rcases Function.bijective_iff_has_inverse.mp \u27e8 hinj, hsur \u27e9 with \u27e8 g', hg' \u27e9\n\n  have bH : Basis B (ZMod 2) H := by\n    constructor\n    exact {\n      toFun := g\n      invFun := g'\n      left_inv := hg'.1\n      right_inv := hg'.2\n      map_add' := AddMonoidHom.map_add _\n      map_smul' := by\n        intro r x\n        rcases ZMod.int_cast_surjective r with \u27e8 n, rfl \u27e9\n        change g ((n : ZMod 2) \u2022 x) = (n : ZMod 2) \u2022 g x\n        rw [intCast_smul, intCast_smul]\n        exact AddMonoidHom.map_zsmul g x n\n    }\n  have hH_fin : Fintype H := Module.fintypeOfFintype bH\n  have hH_card : Nat.card H = 2^(FiniteDimensional.finrank \u2124 G) := by\n    rw [Nat.card_eq_fintype_card, Module.card_fintype bH, \u2190 Nat.card_eq_fintype_card (\u03b1 := B), hB_card]\n    congr\n  exact \u27e8 hH_elem, Finite.of_fintype H, hH_card \u27e9\n\nopen QuotientAddGroup\n\n/-- A version of the third isomorphism theorem: if G\u2082 \u2264 G and H' is a subgroup of G\u29f8G\u2082, then there is a canonical isomorphism between H\u29f8H' and G\u29f8N, where N is the preimage of H' in G. A bit clunky; may be a better way to do this -/\nlemma third_iso {G : Type u} [AddCommGroup G] {G\u2082 : AddSubgroup G} (H' : AddSubgroup (G \u29f8 G\u2082)) :\n  let H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  \u2203 e : H \u29f8 H' \u2243+ G \u29f8 N, \u2200 x : G, e (mk' H' (\u03c6 x))= mk' N x := by\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := mk' G\u2082\n  let N := AddSubgroup.comap \u03c6 H'\n  have h1 : G\u2082 \u2264 N := by\n    intro x hx\n    rw [\u2190 eq_zero_iff] at hx\n    have : \u03c6 x = 0 := hx\n    simp [N, this, AddSubgroup.zero_mem H']\n  set H'' := AddSubgroup.map (mk' G\u2082) N\n  have h2 : H' = H'' := by\n    change H' = AddSubgroup.map (mk' G\u2082) N\n    rw [AddSubgroup.map_comap_eq, AddMonoidHom.range_top_of_surjective _ (mk'_surjective G\u2082)]\n    simp\n  let e1 : H \u29f8 H'' \u2243+ G \u29f8 N := quotientQuotientEquivQuotient _ _ h1\n  let e2 := quotientAddEquivOfEq h2\n  set e := e2.trans e1\n  use e\n  intro x\n  convert (quotientQuotientEquivQuotientAux_mk_mk _ _ h1 x) using 1\n\nlemma single {\u03a9 : Type u} [MeasurableSpace \u03a9] [DiscreteMeasurableSpace \u03a9] (\u03bc : Measure \u03a9)\n    [IsProbabilityMeasure \u03bc] {A : Set \u03a9} {z : \u03a9} (hA : \u03bc.real A = 1) (hz : \u03bc.real {z} > 0) :\n    z \u2208 A := by\n  contrapose! hz\n  have : Disjoint {z} A := by simp [hz]\n  replace this := measureReal_union (\u03bc := \u03bc) this (measurableSet_discrete _)\n  simp [hA] at this\n  have h := measureReal_mono (\u03bc := \u03bc) (show insert z A \u2286 Set.univ by simp)\n  simp [this] at h\n  assumption\n\n/-- Given two non-empty finite subsets A, B of a rank n free Z-module G, there exists a subgroup N and points x, y in G/N such that the fibers Ax, By of A, B over x, y respectively are non-empty, one has the inequality\n$$ \\log \\frac{|A| |B|}{|A_x| |B_y|} \u2264 34 (d[U_A; U_B] - d[U_{A_x}; U_{B_y}])$$\nand one has the dimension bound\n$$ n \\log 2 \u2264 \\log |G/N| + 40 d[U_A; U_B].$$\n -/\nlemma weak_PFR_asymm_prelim (A B : Set G) [Finite A] [Finite B] (hnA : A.Nonempty) (hnB : B.Nonempty):\n    \u2203 (N : AddSubgroup G) (x y : G \u29f8 N) (Ax By : Set G), Ax.Nonempty \u2227 By.Nonempty \u2227\n    Set.Finite Ax \u2227 Set.Finite By \u2227 Ax = {z:G | z \u2208 A \u2227 QuotientAddGroup.mk' N z = x } \u2227\n    By = {z:G | z \u2208 B \u2227 QuotientAddGroup.mk' N z = y } \u2227\n    (log 2) * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) +\n      40 * d\u1d64[ A # B ] \u2227 log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By)\n      \u2264 34 * (d\u1d64[ A # B ] - d\u1d64[ Ax # By ]) := by\n  have : Nonempty A := hnA.to_subtype\n  have : Nonempty B := hnB.to_subtype\n  obtain \u27e8 h_elem, h_finite, h_card \u27e9 := weak_PFR_quotient_prelim (G := G)\n  set \u03c8 : G \u2192+ G := zsmulAddGroupHom 2\n  set G\u2082 := AddMonoidHom.range \u03c8\n  set H := G \u29f8 G\u2082\n  let \u03c6 : G \u2192+ H := QuotientAddGroup.mk' G\u2082\n  let _mH : MeasurableSpace H := \u22a4\n  have h_fintype : Fintype H := Fintype.ofFinite H\n  have h_torsionfree := torsion_free (G := G)\n\n  obtain \u27e8 \u03a9, m\u03a9, UA, h\u03bc, hUA_mes, hUA_unif, hUA_mem, hUA_fin \u27e9 := exists_isUniform_measureSpace' A\n  obtain \u27e8 \u03a9', m\u03a9', UB, h\u03bc', hUB_mes, hUB_unif, hUB_mem, hUB_fin \u27e9 :=\n    exists_isUniform_measureSpace' B\n\n  rcases (PFR_projection (\u03c6.toFun \u2218 UA) (\u03c6.toFun \u2218 UB) \u2119 \u2119 (by measurability) (by measurability)) with \u27e8H', \u27e8 hH1, hH2 \u27e9 \u27e9\n  let N := AddSubgroup.comap \u03c6 H'\n  set \u03c6' := QuotientAddGroup.mk' N\n  have _cGN : Countable (G \u29f8 N) := Function.Surjective.countable (QuotientAddGroup.mk'_surjective N)\n  have _msGN : MeasurableSingletonClass (G \u29f8 N) := by\n    constructor\n    intro x\n    exact MeasurableSpace.map_def.mpr (measurableSet_discrete _)\n\n  rcases third_iso H' with \u27e8 e : H \u29f8 H' \u2243+ G \u29f8 N, he \u27e9\n  rcases single_fibres \u03c6' hnA hnB hUA_mes hUB_mes hUA_unif hUB_unif hUA_mem hUB_mem with\n    \u27e8x, y, Ax, By, hAx, hBy, hnAx, hnBy, hcard_ineq\u27e9\n\n  have : Nonempty Ax := hnAx.to_subtype\n  have : Nonempty By := hnBy.to_subtype\n  have Axf : Finite Ax := by rw [hAx]; infer_instance\n  have Byf : Finite By := by rw [hBy]; infer_instance\n\n  have h1 := torsion_dist_shrinking UA UB \u2119 \u2119 hUA_mes hUB_mes h_torsionfree \u03c6\n  have h2 := torsion_dist_shrinking UB UA \u2119 \u2119 hUB_mes hUA_mes h_torsionfree \u03c6\n  rw [rdist_symm] at h2\n  rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes] at h1 h2\n  -- using explicit .toFun casts as this saves a lot of heartbeats\n  change H[\u03c6.toFun \u2218 UA] \u2264 10 * d\u1d64[A # B] at h1\n  change H[\u03c6.toFun \u2218 UB] \u2264 10 * d\u1d64[A # B] at h2\n  replace hH1 : log (Nat.card H') \u2264 40  * d\u1d64[A # B] := by\n    apply hH1.trans\n    linarith\n  replace h_card : log 2 * FiniteDimensional.finrank \u2124 G \u2264 log (Nat.card (G \u29f8 N)) + 40 * d\u1d64[A # B] := by\n    rw [mul_comm, \u2190 log_rpow (by norm_num)]\n    norm_cast\n    classical\n    rwa [\u2190 h_card, \u2190 Nat.card_congr e.toEquiv, Nat.card_eq_fintype_card, Nat.card_eq_fintype_card, \u2190 AddSubgroup.index_mul_card H', AddSubgroup.index_eq_card, Nat.cast_mul, log_mul, add_le_add_iff_left, \u2190 Nat.card_eq_fintype_card]\n    all_goals norm_cast; exact Fintype.card_ne_zero\n\n\n  use N, x, y, Ax, By\n  refine \u27e8 hnAx, hnBy, Ax.toFinite, By.toFinite, hAx, hBy, h_card, ?_ \u27e9\n\n  replace hH2 : H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] \u2264 34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] := by\n    set X := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UA\n    set Y := ((mk' H').toFun \u2218 \u03c6.toFun) \u2218 UB\n    have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n    have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n    change H[X] + H[Y] \u2264 34 * d[X # Y] at hH2\n\n    have ha : \u03c6'.toFun \u2218 UA = e.toFun \u2218 X := by ext x; exact (he (UA x)).symm\n    have hb : \u03c6'.toFun \u2218 UB = e.toFun \u2218 Y := by ext x; exact (he (UB x)).symm\n    have he_inj : Function.Injective e.toFun := AddEquiv.injective e\n    rw [ha, hb, entropy_comp_of_injective _ hX _ he_inj, entropy_comp_of_injective _ hY _ he_inj]\n    have : d[e.toFun \u2218 X # e.toFun \u2218 Y] = d[X # Y] :=  rdist_of_inj hX hY e.toAddMonoidHom he_inj\n    rwa [this]\n\n  set X : \u03a9 \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UA\n  set Y : \u03a9' \u2192 G \u29f8 N := \u03c6'.toFun \u2218 UB\n  have hX : Measurable X := Measurable.comp (measurable_discrete _) hUA_mes\n  have hY : Measurable Y := Measurable.comp (measurable_discrete _) hUB_mes\n  rcases le_iff_lt_or_eq.mp (rdist_nonneg (\u03bc := \u2119) (\u03bc' := \u2119) hX hY) with h | h\n  swap\n  . rw [\u2190 h] at hH2\n    have hH2A : H[X] \u2265 0 := entropy_nonneg _ _\n    have hH2B : H[Y] \u2265 0 := entropy_nonneg _ _\n    have hH2A' : H[X] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n    have hH2B' : H[Y] \u2264 0 := by linarith only [hH2, hH2A, hH2B]\n\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hX hH2A' with \u27e8 x', hx \u27e9\n    rcases const_of_nonpos_entropy (\u03bc := \u2119) hY hH2B' with \u27e8 y', hy \u27e9\n\n    have hAAx {z : G} (hz : z \u2208 A) : \u03c6'.toFun z = x' := by\n      change (\u2119).real (UA\u207b\u00b9' (\u03c6'\u207b\u00b9' {x'})) = 1 at hx\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUA_mes (measurableSet_discrete _)] at hx\n      set Af := A.toFinite.toFinset\n      have hUAf : IsUniform Af UA := by\n        convert hUA_unif; simp only [Af, Set.Finite.coe_toFinset]\n      have hnAf : 0 < Nat.card Af := by simp only [Af, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Af := by simp [Af, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UA \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUAf hUA_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UA) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUA_mes)\n      replace this := single ((\u2119).map UA) hx this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hxx : Ax = A := by\n      have h : hnAx.some \u2208 Ax := hnAx.some_mem\n      simp [hAx] at h \u22a2\n      have := hAAx h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hAAx hz\n\n    have hBBy {z : G} (hz : z \u2208 B) : \u03c6'.toFun z = y' := by\n      change (\u2119).real (UB\u207b\u00b9' (\u03c6'\u207b\u00b9' {y'})) = 1 at hy\n      rw [\u2190 MeasureTheory.map_measureReal_apply hUB_mes (measurableSet_discrete _)] at hy\n      set Bf := B.toFinite.toFinset\n      have hUBf : IsUniform Bf UB := by convert hUB_unif; simp only [Bf, Set.Finite.coe_toFinset]\n      have hnBf : 0 < Nat.card Bf := by simp only [Bf, Set.Finite.mem_toFinset, Nat.card_pos]\n      have hzf : z \u2208 Bf := by simp [Bf, Set.Finite.mem_toFinset, hz]\n      have : (Measure.map UB \u2119).real {z} > 0 := by\n        rw [IsUniform.measureReal_preimage_of_mem' hUBf hUB_mes hzf]\n        positivity\n      have _ : IsProbabilityMeasure ((\u2119).map UB) := by\n        exact MeasureTheory.isProbabilityMeasure_map (Measurable.aemeasurable hUB_mes)\n      replace this := single ((\u2119).map UB) hy this\n      rwa [Set.mem_preimage, Set.mem_singleton_iff] at this\n\n    have hyy : By = B := by\n      have h : hnBy.some \u2208 By := hnBy.some_mem\n      simp [hBy] at h \u22a2\n      have := hBBy h.1\n      simp [h.2] at this\n      intro z hz\n      simp [this]\n      convert hBBy hz\n\n    simp [hxx, hyy]\n\n  have := calc d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (log (Nat.card A) + log (Nat.card B) - log (Nat.card Ax) - log (Nat.card By))\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By))) := by\n      congr\n      convert (four_logs ?_ ?_ ?_ ?_).symm\n      all_goals norm_cast; exact Nat.card_pos\n    _ \u2264 (H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := hcard_ineq\n    _ \u2264 (34 * d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB]) * (d[UA # UB] - d\u1d64[Ax # By]) := by\n      apply mul_le_mul_of_nonneg_right hH2\n      have := rdist_le_avg_ent (Measurable.comp (measurable_discrete \u03c6'.toFun) hUA_mes) (Measurable.comp (measurable_discrete \u03c6'.toFun) hUB_mes)\n      replace this : 0 < H[\u03c6'.toFun \u2218 UA] + H[\u03c6'.toFun \u2218 UB] := by linarith\n      rw [\u2190 mul_le_mul_left this]\n      apply le_trans _ hcard_ineq\n      rw [mul_zero]\n      change 0 \u2264 d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * log ((Nat.card A) * (Nat.card B) / ((Nat.card Ax) * (Nat.card By)))\n      rw [\u2190 mul_zero d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB], mul_le_mul_left h]\n      apply Real.log_nonneg\n      rw [one_le_div]\n      gcongr\n      . apply Nat.card_mono\n        . exact Set.toFinite A\n        rw [hAx]; exact Set.inter_subset_left _ _\n      apply Nat.card_mono\n      . exact Set.toFinite B\n      rw [hBy]; exact Set.inter_subset_left _ _\n      norm_cast\n      exact mul_pos Nat.card_pos Nat.card_pos\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d[UA # UB] - d\u1d64[Ax # By])) := by ring\n    _ = d[\u03c6'.toFun \u2218 UA # \u03c6'.toFun \u2218 UB] * (34 * (d\u1d64[A # B] - d\u1d64[Ax # By])) := by\n      rw [<- rdist_set_eq_rdist h\u03bc h\u03bc' hUA_unif hUB_unif hUA_mes hUB_mes]\n  exact (mul_le_mul_left h).mp this\n\n/-- Separating out the conclusion of `weak_PFR_asymm` for convenience of induction arguments.-/\ndef WeakPFRAsymmConclusion (A B : Set G) : Prop :=\n  \u2203 A' B' : Set G, A' \u2286 A \u2227 B' \u2286 B \u2227 A'.Nonempty \u2227 B'.Nonempty \u2227\n  log (((Nat.card A) * (Nat.card B)) / ((Nat.card A') * (Nat.card B'))) \u2264 34 * d\u1d64[A # B] \u2227\n  max (dimension A') (dimension B') \u2264 (40 / log 2) * d\u1d64[A # B]\n\n/-- The property of two sets A,B of a group G not being contained in cosets of the same proper subgroup -/\ndef not_in_coset {G: Type u} [AddCommGroup G] (A B : Set G) : Prop := AddSubgroup.closure ((A-A) \u222a (B-B)) = \u22a4\n\n\n/-- In fact one has equality here, but this is tricker to prove and not needed for the argument. -/\nlemma dimension_of_shift {G: Type u} [AddCommGroup G]\n  {H: AddSubgroup G} (A : Set H) (x : G) :\n  dimension ((fun a:H \u21a6 (a:G) + x) '' A) \u2264 dimension A := by\n  classical\n  rcases Nat.find_spec (exists_coset_cover A) with \u27e8 S, v, hrank, hshift \u27e9\n  change FiniteDimensional.finrank \u2124 S = dimension A at hrank\n  rw [\u2190 hrank]\n  convert dimension_le_of_coset_cover _ (Submodule.map H.subtype.toIntLinearMap S) (x+v) ?_\n  . apply LinearEquiv.finrank_eq\n    exact Submodule.equivMapOfInjective _ (by simpa using Subtype.val_injective) _\n  intro a ha\n  rw [Set.mem_image] at ha\n  rcases ha with \u27e8 b, \u27e8 hb, hb'\u27e9 \u27e9\n  rw [Submodule.mem_map]\n  use b - v, hshift b hb\n  simp [\u2190 hb']\n  abel\n\nlemma conclusion_transfers {A B : Set G}\n    (G': AddSubgroup G) (A' B' : Set G')\n    (hA : IsShift A A') (hB : IsShift B B') [Finite A'] [Finite B'] (hA' : A'.Nonempty) (hB' : B'.Nonempty)\n    (h : WeakPFRAsymmConclusion A' B') : WeakPFRAsymmConclusion A B := by\n  have : Nonempty A' := hA'.to_subtype\n  have : Nonempty B' := hB'.to_subtype\n  rcases h with \u27e8A'', B'', hA'', hB'', hA''_non, hB''_non, hcard_ineq, hdim_ineq\u27e9\n  rcases hA with \u27e8 x, hA \u27e9\n  set f : G' \u2192 G := fun a \u21a6 (a : G) + x\n  have hf : Function.Injective f := by\n    intro y z hyz\n    simp [f] at hyz\n    exact hyz\n  have hA' : A = f '' A' := by\n    simp_rw [hA, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, f, add_comm]; rfl\n  rcases hB with \u27e8 y, hB \u27e9\n  set g : G' \u2192 G := fun a \u21a6 (a : G) + y\n  have hg : Function.Injective g := by\n    intro y z hyz\n    simp [g] at hyz\n    exact hyz\n  have hB' : B = g '' B' := by\n    simp_rw [hB, \u2190 Set.image_vadd, Set.image_image, vadd_eq_add, g, add_comm]; rfl\n  use f '' A'', g '' B''\n  have : d\u1d64[A # B] = d\u1d64[A' # B'] := by\n    rw [<-rdist_set_of_inj _ _ (\u03c6 := G'.subtype) Subtype.val_injective, <-rdist_set_add_const (G'.subtype '' A') (G'.subtype '' B') x y]\n    congr\n    . rw [hA]\n      ext y\n      simp [Set.mem_vadd_set]\n      constructor\n      . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n        have : x + z + -x \u2208 G' := by simp [w]\n        use this\n        simp\n        convert hw\n      rintro \u27e8 h, ha \u27e9\n      use y + -x\n      constructor\n      . use h\n      abel\n    rw [hB]\n    ext x\n    simp [Set.mem_vadd_set]\n    constructor\n    . rintro \u27e8 z, \u27e8 \u27e8 w, hw \u27e9, rfl \u27e9 \u27e9\n      have : y + z + -y \u2208 G' := by simp [w]\n      use this\n      simp\n      convert hw\n    rintro \u27e8 h, ha \u27e9\n    use x + -y\n    constructor\n    . use h\n    abel\n\n\n  refine \u27e8 ?_, ?_, ?_, ?_, ?_, ?_ \u27e9\n  . simp [hA', hf, hA'']\n  . simp [hB', hg, hB'']\n  . simp [hA''_non]\n  . simp [hB''_non]\n  . convert hcard_ineq using 2\n    . congr 3\n      . rw [hA', Nat.card_image_of_injective hf]\n      . rw [hB', Nat.card_image_of_injective hg]\n      . rw [Nat.card_image_of_injective hf]\n      rw [Nat.card_image_of_injective hg]\n  convert LE.le.trans _ hdim_ineq using 2\n  norm_cast\n  apply max_le_max\n  . exact dimension_of_shift A'' x\n  \u00b7 exact dimension_of_shift B'' y\n\n/-- If $A,B\\subseteq \\mathbb{Z}^d$ are finite non-empty sets then there exist non-empty $A'\\subseteq A$ and $B'\\subseteq B$ such that\n\\[\\log\\frac{\\lvert A\\rvert\\lvert B\\rvert}{\\lvert A'\\rvert\\lvert B'\\rvert}\\leq 34 d[U_A;U_B]\\]\nsuch that $\\max(\\dim A',\\dim B')\\leq \\frac{40}{\\log 2} d[U_A;U_B]$. -/\nlemma weak_PFR_asymm (A B : Set G) [Finite A] [Finite B] (hA : A.Nonempty) (hB : B.Nonempty) : WeakPFRAsymmConclusion A B  := by\n  let P : \u2115 \u2192 Prop := fun M \u21a6 (\u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G) (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G) (_hG_sing: MeasurableSingletonClass G) (A B: Set G) (_hA_fin: Finite A) (_hB_fin: Finite B) (_hA_non: A.Nonempty) (_hB_non: B.Nonempty) (_hM : (Nat.card A) + (Nat.card B) \u2264 M), WeakPFRAsymmConclusion A B)\n  suffices \u2200 M, (\u2200 M', M' < M \u2192 P M') \u2192 P M by\n    set M := (Nat.card A) + (Nat.card B)\n    have hM : (Nat.card A) + (Nat.card B) \u2264 M := Nat.le_refl _\n    convert (Nat.strong_induction_on (p := P) M this) G \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a _ \u2039_\u203a A B \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a hM\n  intro M h_induct\n  -- wlog we can assume A, B are not in cosets of a smaller subgroup\n  suffices \u2200 (G : Type u) (hG_comm : AddCommGroup G) (_hG_free : Module.Free \u2124 G)\n    (_hG_fin : Module.Finite \u2124 G) (_hG_count : Countable G) (hG_mes : MeasurableSpace G)\n    (_hG_sing : MeasurableSingletonClass G) (A B : Set G) (_hA_fin : Finite A) (_hB_fin : Finite B) (_hA_non : A.Nonempty) (_hB_non : B.Nonempty) (_hM : Nat.card A + Nat.card B \u2264 M)\n    (_hnot : NotInCoset A B), WeakPFRAsymmConclusion A B by\n    intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM\n    obtain \u27e8 G', A', B', hAA', hBB', hnot' \u27e9 := wlog_notInCoset hA_non hB_non\n    have hG'_fin : Module.Finite \u2124 G' :=\n      Module.Finite.iff_fg (N := AddSubgroup.toIntSubmodule G').2 (IsNoetherian.noetherian _)\n\n    have hG'_free : Module.Free \u2124 G' := by\n      rcases Submodule.nonempty_basis_of_pid (Module.Free.chooseBasis \u2124 G) (AddSubgroup.toIntSubmodule G') with \u27e8 n, \u27e8 b \u27e9 \u27e9\n      exact Module.Free.of_basis b\n    have hAA'_card: Nat.card A = Nat.card A' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hAA'.card_congr\n    have hBB'_card: Nat.card B = Nat.card B' := (Nat.card_image_of_injective Subtype.val_injective _) \u25b8 hBB'.card_congr\n    have hA_non' : Nonempty A := Set.nonempty_coe_sort.mpr hA_non\n    have hB_non' : Nonempty B := Set.nonempty_coe_sort.mpr hB_non\n\n    rw [hAA'_card, hBB'_card] at hM\n\n    have hA'_nonfin : A'.Nonempty \u2227 Finite A' := by\n      have := Nat.card_pos (\u03b1 := A)\n      rw [hAA'_card, Nat.card_pos_iff] at this\n      exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n    have hB'_nonfin : B'.Nonempty \u2227 Finite B' := by\n      have := Nat.card_pos (\u03b1 := B)\n      rw [hBB'_card, Nat.card_pos_iff] at this\n      exact \u27e8@nonempty_of_nonempty_subtype _ _ this.1, this.2\u27e9\n    obtain \u27e8 hA'_non, hA'_fin \u27e9 := hA'_nonfin\n    obtain \u27e8 hB'_non, hB'_fin \u27e9 := hB'_nonfin\n\n    replace this := this G' _ hG'_free hG'_fin (by infer_instance) (by infer_instance) (by infer_instance) A' B' hA'_fin hB'_fin hA'_non hB'_non hM hnot'\n    exact conclusion_transfers G' A' B' hAA' hBB' hA'_non hB'_non this\n  intro G hG_comm hG_free hG_fin hG_count hG_mes hG_sing A B hA_fin hB_fin hA_non hB_non hM hnot\n  rcases weak_PFR_asymm_prelim A B hA_non hB_non with \u27e8 N, x, y, Ax, By, hAx_non, hBy_non, hAx_fin, hBy_fin, hAx, hBy, hdim, hcard\u27e9\n  have hAxA : Ax \u2286 A := by rw [hAx]; simp\n  have hByB : By \u2286 B := by rw [hBy]; simp\n  have hA_pos : (0 : \u211d) < Nat.card A := Nat.cast_pos.mpr (@Nat.card_pos _ hA_non.to_subtype _)\n  have hB_pos : (0 : \u211d) < Nat.card B := Nat.cast_pos.mpr (@Nat.card_pos _ hB_non.to_subtype _)\n\n  rcases lt_or_ge (Nat.card Ax + Nat.card By) (Nat.card A + Nat.card B) with h | h\n  . replace h := h_induct (Nat.card Ax + Nat.card By) (h.trans_le hM) G hG_comm hG_free hG_fin hG_count hG_mes hG_sing Ax By (Set.finite_coe_iff.mpr hAx_fin) (Set.finite_coe_iff.mpr hBy_fin) hAx_non hBy_non (Eq.le rfl)\n    rcases h with \u27e8 A', B', hA', hB', hA'_non, hB'_non, hcard_ineq, hdim_ineq \u27e9\n    use A', B'\n    have hAx_fin' := Set.finite_coe_iff.mpr hAx_fin\n    have hBy_fin' := Set.finite_coe_iff.mpr hBy_fin\n    have hA'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hAx_fin hA')\n    have hB'_fin' := Set.finite_coe_iff.mpr (Set.Finite.subset hBy_fin hB')\n    have hAx_non' := Set.nonempty_coe_sort.mpr hAx_non\n    have hBy_non' := Set.nonempty_coe_sort.mpr hBy_non\n    have hA'_non' := Set.nonempty_coe_sort.mpr hA'_non\n    have hB'_non' := Set.nonempty_coe_sort.mpr hB'_non\n    have hAx_pos : (0 : \u211d) < Nat.card Ax := Nat.cast_pos.mpr Nat.card_pos\n    have hBy_pos : (0 : \u211d) < Nat.card By := Nat.cast_pos.mpr Nat.card_pos\n    have hA'_pos : (0 : \u211d) < Nat.card A' := Nat.cast_pos.mpr Nat.card_pos\n    have hB'_pos : (0 : \u211d) < Nat.card B' := Nat.cast_pos.mpr Nat.card_pos\n    have hAxA_le : (Nat.card Ax : \u211d) \u2264 (Nat.card A : \u211d) := Nat.cast_le.mpr (Nat.card_mono A.toFinite hAxA)\n    have hByB_le : (Nat.card By : \u211d) \u2264 (Nat.card B : \u211d) := Nat.cast_le.mpr (Nat.card_mono B.toFinite hByB)\n\n    refine \u27e8 hA'.trans hAxA, hB'.trans hByB, hA'_non, hB'_non, ?_, ?_ \u27e9\n    . rw [four_logs hA_pos hB_pos hA'_pos hB'_pos]\n      rw [four_logs hAx_pos hBy_pos hA'_pos hB'_pos] at hcard_ineq\n      linarith only [hcard, hcard_ineq]\n    apply hdim_ineq.trans\n    gcongr\n    linarith only [Real.log_le_log hAx_pos hAxA_le, Real.log_le_log hBy_pos hByB_le, hcard]\n  use A, B\n  refine \u27e8 Eq.subset rfl, Eq.subset rfl, hA_non, hB_non, ?_, ?_ \u27e9\n  . have := hA_non.to_subtype\n    have := hB_non.to_subtype\n    apply LE.le.trans _ <| mul_nonneg (by norm_num) <| rdist_set_nonneg A B\n    rw [div_self (by positivity)]\n    simp\n  have hAx_eq : Ax = A := by\n    apply Set.Finite.eq_of_subset_of_card_le A.toFinite hAxA\n    linarith only [h, Nat.card_mono B.toFinite hByB]\n  have hBy_eq : By = B := by\n    apply Set.Finite.eq_of_subset_of_card_le B.toFinite hByB\n    linarith only [h, Nat.card_mono A.toFinite hAxA]\n  have hN : N = \u22a4 := by\n    have : (A-A) \u222a (B-B) \u2286 N := by\n      rw [\u2190 hAx_eq, \u2190 hBy_eq, hAx, hBy]\n      intro z hz\n      simp [Set.mem_sub] at hz\n      convert (QuotientAddGroup.eq_zero_iff z).mp ?_\n      . infer_instance\n      rcases hz with \u27e8 a, \u27e8 -, ha\u27e9, a', \u27e8-, ha'\u27e9, haa' \u27e9 | \u27e8 b, \u27e8 -, hb\u27e9, b', \u27e8 -,hb'\u27e9, hbb' \u27e9\n      . rw [\u2190 haa']; simp [ha, ha']\n      rw [\u2190 hbb']; simp [hb, hb']\n    rw [\u2190 AddSubgroup.closure_le, hnot] at this\n    exact top_le_iff.mp this\n  have : Nat.card (G \u29f8 N) = 1 := by\n    rw [Nat.card_eq_one_iff_unique]\n    constructor\n    . rw [hN]\n      exact  QuotientAddGroup.subsingleton_quotient_top\n    infer_instance\n  simp [this] at hdim\n  rw [\u2190 le_div_iff' (by positivity)] at hdim\n  convert LE.le.trans ?_ hdim using 1\n  . field_simp\n  simp\n  exact \u27e8 dimension_le_rank A, dimension_le_rank B \u27e9\n\n/-- If $A\\subseteq \\mathbb{Z}^d$ is a finite non-empty set with $d[U_A;U_A]\\leq \\log K$ then there exists a non-empty $A'\\subseteq A$ such that\n$\\lvert A'\\rvert\\geq K^{-17}\\lvert A\\rvert$\nand $\\dim A'\\leq \\frac{40}{\\log 2} \\log K$. -/\nlemma weak_PFR {A : Set G} [Finite A] {K : \u211d} (hA : A.Nonempty) (hK: 0 < K) (hdist: d\u1d64[A # A] \u2264 log K):\n    \u2203 A' : Set G, A' \u2286 A \u2227 (Nat.card A') \u2265 K^(-17 : \u211d) * (Nat.card A)\n    \u2227 (dimension A') \u2264 (40 / log 2) * log K := by\n  rcases weak_PFR_asymm A A hA hA with \u27e8A', A'', hA', hA'', hA'nonempty, hA''nonempty, hcard, hdim\u27e9\n\n  have : \u2203 B : Set G, B \u2286 A \u2227 (Nat.card B) \u2265 (Nat.card A') \u2227 (Nat.card B) \u2265 (Nat.card A'') \u2227 (dimension B) \u2264\nmax (dimension A') (dimension A'') := by\n    rcases lt_or_ge (Nat.card A') (Nat.card A'') with h | h\n    . use A''\n      exact \u27e8hA'', by linarith, by linarith, le_max_right _ _\u27e9\n    use A'\n    exact \u27e8hA', by linarith, by linarith, le_max_left _ _\u27e9\n\n  rcases this with \u27e8B, hB, hBcard, hBcard', hBdim\u27e9\n  use B\n  have hApos : Nat.card A > 0 := by\n    rw [gt_iff_lt, Nat.card_pos_iff]\n    exact \u27e8hA.to_subtype, inferInstance\u27e9\n  have hA'pos : Nat.card A' > 0 := by\n    rw [gt_iff_lt, Nat.card_pos_iff]\n    refine \u27e8 hA'nonempty.to_subtype, Finite.Set.subset _ hA' \u27e9\n  have hA''pos : Nat.card A'' > 0 := by\n    rw [gt_iff_lt, Nat.card_pos_iff]\n    refine \u27e8 hA''nonempty.to_subtype, Finite.Set.subset _ hA'' \u27e9\n  have hBpos : Nat.card B > 0 := by linarith\n\n  refine \u27e8hB, ?_, ?_\u27e9\n  . have := calc 2 * log ((Nat.card A) / (Nat.card B))\n      _ = log ( ((Nat.card A) * (Nat.card A)) / ((Nat.card B) * (Nat.card B)) ) := by\n        convert (log_pow (((Nat.card A) : \u211d)/(Nat.card B)) 2).symm\n        field_simp\n        rw [\u2190 pow_two, \u2190 pow_two]\n      _ \u2264 log ( ((Nat.card A) * (Nat.card A)) / ((Nat.card A') * (Nat.card A'')) ) := by\n        apply log_le_log\n        . positivity\n        gcongr\n      _ \u2264 34 * d\u1d64[A # A] := hcard\n      _ \u2264 34 * log K := mul_le_mul_of_nonneg_left hdist (by linarith)\n      _ = 2 * (17 * log K) := by ring\n      _ = 2 * log (K^17) := by\n        congr\n        convert (log_pow K 17).symm\n    rw [mul_le_mul_left (by norm_num), log_le_log_iff (by positivity) (by positivity), div_le_iff (by positivity), <- mul_inv_le_iff (by positivity), <-ge_iff_le, mul_comm] at this\n    convert this using 2\n    convert zpow_neg K 17 using 1\n    norm_cast\n  calc (dimension B : \u211d)\n    _ \u2264 (((max (dimension A') (dimension A'')) : \u2115) : \u211d) := by norm_cast\n    _ \u2264 (40 / log 2) * d\u1d64[A # A] := hdim\n    _ \u2264 (40 / log 2) * log K := mul_le_mul_of_nonneg_left hdist (by positivity)\n\n", "theoremStatement": "/-- Let $A\\subseteq \\mathbb{Z}^d$ and $\\lvert A-A\\rvert\\leq K\\lvert A\\rvert$.\nThere exists $A'\\subseteq A$ such that $\\lvert A'\\rvert \\geq K^{-17}\\lvert A\\rvert$\nand $\\dim A' \\leq \\frac{40}{\\log 2} \\log K$.-/\ntheorem weak_PFR_int {A : Set G} [Finite A] (hnA : A.Nonempty) {K : \u211d} (hK : 0 < K)\n    (hA: Nat.card (A-A) \u2264 K * Nat.card A) :\n    \u2203 A' : Set G, A' \u2286 A \u2227 Nat.card A' \u2265 K ^ (-17 : \u211d) * (Nat.card A) \u2227\n      dimension A' \u2264 (40 / log 2) * log K ", "theoremName": "weak_PFR_int", "fileCreated": {"commit": "25309ad", "date": "2023-12-11"}, "theoremCreated": {"commit": "c0c547a", "date": "2024-01-16"}, "file": "PFR/PFR/WeakPFR.lean", "module": "PFR.WeakPFR", "jsonFile": "PFR.WeakPFR.jsonl", "positionMetadata": {"lineInFile": 1228, "tokenPositionInFile": 63501, "theoremPositionInFile": 30}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 2, "repositoryPremises": true, "numRepositoryPremises": 4, "numPremises": 243, "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", 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"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", 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"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.NeZero", "Mathlib.Algebra.GroupWithZero.NeZero", "Mathlib.Init.Order.LinearOrder", "Mathlib.Order.Notation", "Mathlib.Order.Basic", "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.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.Order.Synonym", "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.Sub.Defs", "Mathlib.Algebra.Order.Group.Defs", "Mathlib.Algebra.Order.Monoid.Canonical.Defs", "Mathlib.Algebra.Order.Monoid.MinMax", "Mathlib.Algebra.Order.ZeroLEOne", "Mathlib.Data.Nat.Cast.Defs", "Mathlib.Algebra.Order.Monoid.NatCast", "Mathlib.Algebra.Order.Ring.Lemmas", "Mathlib.Data.Int.Cast.Defs", "Mathlib.Algebra.Ring.Defs", "Mathlib.Algebra.Order.Ring.Defs", "Mathlib.Algebra.Quotient", "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.Data.Finite.Defs", "Mathlib.Algebra.Group.TypeTags", "Mathlib.Algebra.Group.Equiv.TypeTags", "Mathlib.Algebra.CharZero.Defs", "Mathlib.Algebra.Order.Ring.CharZero", "Mathlib.Algebra.Group.Int", "Mathlib.Data.Int.Cast.Basic", "Mathlib.Algebra.Ring.Int", "Mathlib.Algebra.Group.OrderSynonym", "Mathlib.Algebra.Order.Monoid.OrderDual", "Mathlib.Tactic.Monotonicity.Attr", "Mathlib.Algebra.GroupPower.CovariantClass", "Mathlib.Algebra.Group.Units.Hom", "Mathlib.Algebra.Group.Units.Equiv", "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.Divisibility.Basic", "Mathlib.Algebra.Order.Group.Int", "Mathlib.Algebra.Order.Ring.Int", "Mathlib.Algebra.Group.InjSurj", "Mathlib.Algebra.GroupWithZero.InjSurj", "Mathlib.Algebra.GroupWithZero.Hom", "Mathlib.Algebra.Opposites", "Mathlib.Algebra.Ring.Basic", "Mathlib.Algebra.Ring.Hom.Defs", "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.Algebra.Ring.Hom.Basic", "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.Ring.Semiconj", "Mathlib.Algebra.Ring.InjSurj", "Mathlib.Algebra.Ring.Units", "Mathlib.Data.Bracket", "Mathlib.Algebra.Ring.Commute", "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.Sub.Canonical", "Mathlib.Algebra.Order.Ring.Canonical", "Mathlib.Algebra.Order.Ring.Nat", "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.Set.Intervals.Basic", "Mathlib.Data.Set.Image", "Mathlib.Data.Set.Prod", "Mathlib.Data.Set.NAry", "Mathlib.Order.Directed", "Mathlib.Order.Bounds.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.GroupTheory.GroupAction.Defs", "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.WellFounded", "Mathlib.Logic.Pairwise", "Mathlib.Data.Bool.Set", "Mathlib.Data.Nat.Set", "Mathlib.Control.ULift", "Mathlib.Data.ULift", "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.Order.ConditionallyCompleteLattice.Basic", "Mathlib.Order.LatticeIntervals", "Mathlib.Data.Set.Intervals.Image", "Mathlib.Data.Set.Intervals.UnorderedInterval", "Mathlib.Data.Set.Intervals.OrderEmbedding", "Mathlib.Data.Set.Pairwise.Basic", "Mathlib.Order.Antichain", "Mathlib.Data.Set.Intervals.OrdConnected", "Mathlib.Order.CompleteLatticeIntervals", "Mathlib.Algebra.Order.Nonneg.Ring", "Mathlib.Algebra.Order.Nonneg.Field", "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.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.Algebra.Group.Opposite", "Mathlib.Algebra.Group.Prod", "Mathlib.Order.Antisymmetrization", "Mathlib.Order.Cover", "Mathlib.Algebra.Function.Support", "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.Algebra.Group.Hom.Instances", "Mathlib.Algebra.Group.Pi.Lemmas", "Mathlib.Algebra.Function.Indicator", "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.Piecewise", "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.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.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.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.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.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.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.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.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.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.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.Projection", "Mathlib.LinearAlgebra.Basis.VectorSpace", "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.GroupTheory.SpecificGroups.Cyclic", "Mathlib.GroupTheory.PGroup", "Mathlib.GroupTheory.Torsion", "Mathlib.LinearAlgebra.Dimension.Basic", "Mathlib.LinearAlgebra.Dimension.Finrank", "Mathlib.LinearAlgebra.InvariantBasisNumber", "Mathlib.LinearAlgebra.Dimension.StrongRankCondition", "Mathlib.LinearAlgebra.FreeModule.PID", "PFR.Mathlib.Data.Set.Pointwise.SMul", "Mathlib.Tactic.GCongr", "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.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.Algebra.AddTorsor", "Mathlib.Topology.Algebra.Constructions", "Mathlib.Topology.Algebra.ConstMulAction", "Mathlib.Topology.Algebra.MulAction", "Mathlib.Topology.ContinuousFunction.Basic", "Mathlib.Topology.Algebra.Monoid", "Mathlib.Topology.Algebra.InfiniteSum.Basic", "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.Data.Set.Pointwise.Interval", "Mathlib.Topology.Algebra.Order.Field", "Mathlib.Topology.Order.MonotoneConvergence", "Mathlib.Topology.Algebra.InfiniteSum.Order", "Mathlib.Topology.UniformSpace.AbstractCompletion", "Mathlib.Topology.UniformSpace.Completion", "Mathlib.Topology.Algebra.UniformMulAction", "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.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.Topology.Algebra.Module.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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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.LinearAlgebra.FreeModule.Finite.Basic", "Mathlib.LinearAlgebra.Dimension.Free", "Mathlib.LinearAlgebra.Isomorphisms", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.TauFunctional", "PFR.HundredPercent", "PFR.Fibring", "PFR.FirstEstimate", "PFR.Mathlib.Probability.Independence.FourVariables", "PFR.SecondEstimate", "PFR.Endgame", "PFR.EntropyPFR", "Mathlib.Combinatorics.Additive.RuzsaCovering", "Mathlib.GroupTheory.Complement", "PFR.Mathlib.GroupTheory.Subgroup.Pointwise", "Mathlib.Data.Real.ENatENNReal", "Mathlib.Data.Set.Card", "PFR.ForMathlib.Entropy.RuzsaSetDist", "Mathlib.Tactic.Qify", "Mathlib.Tactic.Rify", "PFR.Tactic.RPowSimp", "PFR.Main", "PFR.ImprovedPFR", "PFR.Mathlib.GroupTheory.Torsion"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  apply weak_PFR hnA hK ((rdist_set_le A A hnA hnA).trans _)\n  suffices log (Nat.card (A-A)) \u2264 log K + log (Nat.card A) by linarith\n  rw [\u2190 log_mul (by positivity) _]\n  . apply log_le_log _ hA\n    norm_cast\n    have : Nonempty (A-A) := by\n      exact Set.Nonempty.coe_sort (Set.Nonempty.sub hnA hnA)\n    apply Nat.card_pos\n  norm_cast\n  apply ne_of_gt (@Nat.card_pos _ hnA.to_subtype _)", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 392}}
{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\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)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n  /-- The first variable in a package. -/\n  X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n  /-- The second variable in a package. -/\n  X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n  hmeas1 : Measurable X\u2080\u2081\n  hmeas2 : Measurable X\u2080\u2082\n  \u03b7 : \u211d\n  h\u03b7 : 0 < \u03b7\n  h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n  d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\n", "theoremStatement": "lemma continuous_tau_restrict_probabilityMeasure\n    [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n    Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) ", "theoremName": "continuous_tau_restrict_probabilityMeasure", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "6854e2a", "date": "2023-11-20"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 72, "tokenPositionInFile": 2703, "theoremPositionInFile": 3}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 10, "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", 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"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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have obs\u2081 : 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  have obs\u2082 : Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n    continuous_rdist_restrict_probabilityMeasure\n  have obs\u2083 : 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  continuity", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 618}}
{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\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)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n  /-- The first variable in a package. -/\n  X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n  /-- The second variable in a package. -/\n  X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n  hmeas1 : Measurable X\u2080\u2081\n  hmeas2 : Measurable X\u2080\u2082\n  \u03b7 : \u211d\n  h\u03b7 : 0 < \u03b7\n  h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n  d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n    [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n    Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n  have obs\u2081 : 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  have obs\u2082 : Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n    continuous_rdist_restrict_probabilityMeasure\n  have obs\u2083 : 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  continuity\n\n", "theoremStatement": "/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n    {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n    {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n    \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] ", "theoremName": "ProbabilityTheory.IdentDistrib.tau_eq", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 87, "tokenPositionInFile": 3541, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 9, "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", 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"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", 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"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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simp only [tau]\n  rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n      (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n      h\u2081.rdist_eq h\u2082]", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 167}}
{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\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)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n  /-- The first variable in a package. -/\n  X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n  /-- The second variable in a package. -/\n  X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n  hmeas1 : Measurable X\u2080\u2081\n  hmeas2 : Measurable X\u2080\u2082\n  \u03b7 : \u211d\n  h\u03b7 : 0 < \u03b7\n  h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n  d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n    [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n    Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n  have obs\u2081 : 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  have obs\u2082 : Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n    continuous_rdist_restrict_probabilityMeasure\n  have obs\u2083 : 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  continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n    {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n    {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n    \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n  simp only [tau]\n  rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n      (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n      h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n  \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n      \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n", "theoremStatement": "/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n    [MeasureSpace \u03a9']\n    {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n    tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p 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"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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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", 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"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", "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", 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"PFR.Mathlib.MeasureTheory.Measure.MeasureSpace", "PFR.ForMathlib.Entropy.Kernel.RuzsaDist", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]", "proofType": "tactic", "proofLengthLines": 1, "proofLengthTokens": 54}}
{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\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)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n  /-- The first variable in a package. -/\n  X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n  /-- The second variable in a package. -/\n  X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n  hmeas1 : Measurable X\u2080\u2081\n  hmeas2 : Measurable X\u2080\u2082\n  \u03b7 : \u211d\n  h\u03b7 : 0 < \u03b7\n  h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n  d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n    [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n    Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n  have obs\u2081 : 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  have obs\u2082 : Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n    continuous_rdist_restrict_probabilityMeasure\n  have obs\u2083 : 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  continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n    {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n    {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n    \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n  simp only [tau]\n  rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n      (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n      h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n  \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n      \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n    [MeasureSpace \u03a9']\n    {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n    tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n  simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n", "theoremStatement": "/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\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 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] ", "theoremName": "tau_min_exists_measure", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "96a97b5", "date": "2023-11-27"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 114, "tokenPositionInFile": 5090, "theoremPositionInFile": 7}, "dependencyMetadata": 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"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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n  have : DiscreteTopology G := \u27e8rfl\u27e9\n  have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n  let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n    fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n  have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n  haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n  obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n                          _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n  use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n  refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n  intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n  rw [isMinOn_univ_iff] at h\u03bc\n  let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n  exact h\u03bc \u03bd", "proofType": "tactic", "proofLengthLines": 15, "proofLengthTokens": 964}}
{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\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)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n  /-- The first variable in a package. -/\n  X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n  /-- The second variable in a package. -/\n  X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n  hmeas1 : Measurable X\u2080\u2081\n  hmeas2 : Measurable X\u2080\u2082\n  \u03b7 : \u211d\n  h\u03b7 : 0 < \u03b7\n  h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n  d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n    [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n    Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n  have obs\u2081 : 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  have obs\u2082 : Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n    continuous_rdist_restrict_probabilityMeasure\n  have obs\u2083 : 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  continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n    {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n    {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n    \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n  simp only [tau]\n  rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n      (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n      h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n  \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n      \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n    [MeasureSpace \u03a9']\n    {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n    tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n  simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\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 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] := by\n  let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n  have : DiscreteTopology G := \u27e8rfl\u27e9\n  have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n  let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n    fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n  have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n  haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n  obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n                          _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n  use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n  refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n  intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n  rw [isMinOn_univ_iff] at h\u03bc\n  let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n  exact h\u03bc \u03bd\n\n", "theoremStatement": "/-- A pair of random variables minimizing $\u03c4$ exists. -/\nlemma tau_minimizer_exists [MeasurableSingletonClass G] :\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\n    tau_minimizes p X\u2081 X\u2082 ", "theoremName": "tau_minimizer_exists", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "13e55a8", "date": "2023-11-28"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 136, "tokenPositionInFile": 6414, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 5, "repositoryPremises": true, "numRepositoryPremises": 7, "numPremises": 50, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", 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"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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  let \u03bc := (tau_min_exists_measure p).choose\n  have : IsProbabilityMeasure \u03bc.1 := (tau_min_exists_measure p).choose_spec.1\n  have : IsProbabilityMeasure \u03bc.2 := (tau_min_exists_measure p).choose_spec.2.1\n  have P : IsProbabilityMeasure (\u03bc.1.prod \u03bc.2) := by infer_instance\n  let M : MeasureSpace (G \u00d7 G) := \u27e8\u03bc.1.prod \u03bc.2\u27e9\n  refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_\u27e9\n  intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n  have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] :=\n    ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n  convert (tau_min_exists_measure p).choose_spec.2.2 \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082", "proofType": "tactic", "proofLengthLines": 10, "proofLengthTokens": 653}}
{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\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)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n  /-- The first variable in a package. -/\n  X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n  /-- The second variable in a package. -/\n  X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n  hmeas1 : Measurable X\u2080\u2081\n  hmeas2 : Measurable X\u2080\u2082\n  \u03b7 : \u211d\n  h\u03b7 : 0 < \u03b7\n  h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n  d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n    [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n    Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n  have obs\u2081 : 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  have obs\u2082 : Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n    continuous_rdist_restrict_probabilityMeasure\n  have obs\u2083 : 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  continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n    {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n    {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n    \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n  simp only [tau]\n  rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n      (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n      h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n  \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n      \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n    [MeasureSpace \u03a9']\n    {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n    tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n  simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\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 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] := by\n  let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n  have : DiscreteTopology G := \u27e8rfl\u27e9\n  have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n  let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n    fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n  have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n  haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n  obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n                          _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n  use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n  refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n  intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n  rw [isMinOn_univ_iff] at h\u03bc\n  let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n  exact h\u03bc \u03bd\n\n/-- A pair of random variables minimizing $\u03c4$ exists. -/\nlemma tau_minimizer_exists [MeasurableSingletonClass G] :\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\n    tau_minimizes p X\u2081 X\u2082 := by\n  let \u03bc := (tau_min_exists_measure p).choose\n  have : IsProbabilityMeasure \u03bc.1 := (tau_min_exists_measure p).choose_spec.1\n  have : IsProbabilityMeasure \u03bc.2 := (tau_min_exists_measure p).choose_spec.2.1\n  have P : IsProbabilityMeasure (\u03bc.1.prod \u03bc.2) := by infer_instance\n  let M : MeasureSpace (G \u00d7 G) := \u27e8\u03bc.1.prod \u03bc.2\u27e9\n  refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_\u27e9\n  intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n  have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] :=\n    ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n  convert (tau_min_exists_measure p).choose_spec.2.2 \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n\n\nvariable [MeasureSpace \u03a9] [h\u03a9\u2081 : MeasureSpace \u03a9'\u2081] [h\u03a9\u2082 : MeasureSpace \u03a9'\u2082]\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2082)]\n  {X\u2081 : \u03a9 \u2192 G} {X\u2082 : \u03a9 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n\n", "theoremStatement": "lemma is_tau_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n    \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[X'\u2081 # X'\u2082 | p] ", "theoremName": "is_tau_min", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "828bcd7", "date": "2023-11-20"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", 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"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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  let \u03bd\u2081 := (\u2119 : Measure \u03a9'\u2081).map X'\u2081\n  let \u03bd\u2082 := (\u2119 : Measure \u03a9'\u2082).map X'\u2082\n  have B : \u03c4[X'\u2081 # X'\u2082 | p] = \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] :=\n    (identDistrib_id_right h1.aemeasurable).tau_eq p (identDistrib_id_right h2.aemeasurable)\n  convert h \u03bd\u2081 \u03bd\u2082 (isProbabilityMeasure_map h1.aemeasurable)\n    (isProbabilityMeasure_map h2.aemeasurable)", "proofType": "tactic", "proofLengthLines": 6, "proofLengthTokens": 340}}
{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\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)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n  /-- The first variable in a package. -/\n  X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n  /-- The second variable in a package. -/\n  X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n  hmeas1 : Measurable X\u2080\u2081\n  hmeas2 : Measurable X\u2080\u2082\n  \u03b7 : \u211d\n  h\u03b7 : 0 < \u03b7\n  h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n  d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n    [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n    Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n  have obs\u2081 : 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  have obs\u2082 : Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n    continuous_rdist_restrict_probabilityMeasure\n  have obs\u2083 : 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  continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n    {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n    {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n    \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n  simp only [tau]\n  rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n      (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n      h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n  \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n      \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n    [MeasureSpace \u03a9']\n    {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n    tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n  simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\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 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] := by\n  let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n  have : DiscreteTopology G := \u27e8rfl\u27e9\n  have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n  let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n    fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n  have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n  haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n  obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n                          _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n  use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n  refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n  intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n  rw [isMinOn_univ_iff] at h\u03bc\n  let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n  exact h\u03bc \u03bd\n\n/-- A pair of random variables minimizing $\u03c4$ exists. -/\nlemma tau_minimizer_exists [MeasurableSingletonClass G] :\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\n    tau_minimizes p X\u2081 X\u2082 := by\n  let \u03bc := (tau_min_exists_measure p).choose\n  have : IsProbabilityMeasure \u03bc.1 := (tau_min_exists_measure p).choose_spec.1\n  have : IsProbabilityMeasure \u03bc.2 := (tau_min_exists_measure p).choose_spec.2.1\n  have P : IsProbabilityMeasure (\u03bc.1.prod \u03bc.2) := by infer_instance\n  let M : MeasureSpace (G \u00d7 G) := \u27e8\u03bc.1.prod \u03bc.2\u27e9\n  refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_\u27e9\n  intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n  have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] :=\n    ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n  convert (tau_min_exists_measure p).choose_spec.2.2 \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n\n\nvariable [MeasureSpace \u03a9] [h\u03a9\u2081 : MeasureSpace \u03a9'\u2081] [h\u03a9\u2082 : MeasureSpace \u03a9'\u2082]\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2082)]\n  {X\u2081 : \u03a9 \u2192 G} {X\u2082 : \u03a9 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n\nlemma is_tau_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n    \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[X'\u2081 # X'\u2082 | p] := by\n  let \u03bd\u2081 := (\u2119 : Measure \u03a9'\u2081).map X'\u2081\n  let \u03bd\u2082 := (\u2119 : Measure \u03a9'\u2082).map X'\u2082\n  have B : \u03c4[X'\u2081 # X'\u2082 | p] = \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] :=\n    (identDistrib_id_right h1.aemeasurable).tau_eq p (identDistrib_id_right h2.aemeasurable)\n  convert h \u03bd\u2081 \u03bd\u2082 (isProbabilityMeasure_map h1.aemeasurable)\n    (isProbabilityMeasure_map h2.aemeasurable)\n\n", "theoremStatement": "/-- Let `X\u2081` and `X\u2082` be tau-minimizers associated to `p`, with $d[X_1,X_2]=k$, then\n$$ d[X'_1;X'_2] \\geq\n    k - \\eta (d[X^0_1;X'_1] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2] - d[X^0_2;X_2] )$$\nfor any $G$-valued random variables $X'_1,X'_2$.\n-/\nlemma distance_ge_of_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n    d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081 # X'\u2081] - d[p.X\u2080\u2081 # X\u2081]) - p.\u03b7 * (d[p.X\u2080\u2082 # X'\u2082] - d[p.X\u2080\u2082 # X\u2082])\n      \u2264 d[X'\u2081 # X'\u2082] ", "theoremName": "distance_ge_of_min", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/TauFunctional.lean", 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"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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have Z := is_tau_min p h h1 h2\n  simp [tau] at Z\n  linarith", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 67}}
{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\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)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n  /-- The first variable in a package. -/\n  X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n  /-- The second variable in a package. -/\n  X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n  hmeas1 : Measurable X\u2080\u2081\n  hmeas2 : Measurable X\u2080\u2082\n  \u03b7 : \u211d\n  h\u03b7 : 0 < \u03b7\n  h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n  d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n    [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n    Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n  have obs\u2081 : 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  have obs\u2082 : Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n    continuous_rdist_restrict_probabilityMeasure\n  have obs\u2083 : 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  continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n    {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n    {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n    \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n  simp only [tau]\n  rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n      (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n      h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n  \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n      \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n    [MeasureSpace \u03a9']\n    {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n    tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n  simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\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 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] := by\n  let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n  have : DiscreteTopology G := \u27e8rfl\u27e9\n  have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n  let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n    fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n  have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n  haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n  obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n                          _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n  use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n  refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n  intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n  rw [isMinOn_univ_iff] at h\u03bc\n  let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n  exact h\u03bc \u03bd\n\n/-- A pair of random variables minimizing $\u03c4$ exists. -/\nlemma tau_minimizer_exists [MeasurableSingletonClass G] :\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\n    tau_minimizes p X\u2081 X\u2082 := by\n  let \u03bc := (tau_min_exists_measure p).choose\n  have : IsProbabilityMeasure \u03bc.1 := (tau_min_exists_measure p).choose_spec.1\n  have : IsProbabilityMeasure \u03bc.2 := (tau_min_exists_measure p).choose_spec.2.1\n  have P : IsProbabilityMeasure (\u03bc.1.prod \u03bc.2) := by infer_instance\n  let M : MeasureSpace (G \u00d7 G) := \u27e8\u03bc.1.prod \u03bc.2\u27e9\n  refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_\u27e9\n  intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n  have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] :=\n    ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n  convert (tau_min_exists_measure p).choose_spec.2.2 \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n\n\nvariable [MeasureSpace \u03a9] [h\u03a9\u2081 : MeasureSpace \u03a9'\u2081] [h\u03a9\u2082 : MeasureSpace \u03a9'\u2082]\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2082)]\n  {X\u2081 : \u03a9 \u2192 G} {X\u2082 : \u03a9 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n\nlemma is_tau_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n    \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[X'\u2081 # X'\u2082 | p] := by\n  let \u03bd\u2081 := (\u2119 : Measure \u03a9'\u2081).map X'\u2081\n  let \u03bd\u2082 := (\u2119 : Measure \u03a9'\u2082).map X'\u2082\n  have B : \u03c4[X'\u2081 # X'\u2082 | p] = \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] :=\n    (identDistrib_id_right h1.aemeasurable).tau_eq p (identDistrib_id_right h2.aemeasurable)\n  convert h \u03bd\u2081 \u03bd\u2082 (isProbabilityMeasure_map h1.aemeasurable)\n    (isProbabilityMeasure_map h2.aemeasurable)\n\n/-- Let `X\u2081` and `X\u2082` be tau-minimizers associated to `p`, with $d[X_1,X_2]=k$, then\n$$ d[X'_1;X'_2] \\geq\n    k - \\eta (d[X^0_1;X'_1] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2] - d[X^0_2;X_2] )$$\nfor any $G$-valued random variables $X'_1,X'_2$.\n-/\nlemma distance_ge_of_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n    d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081 # X'\u2081] - d[p.X\u2080\u2081 # X\u2081]) - p.\u03b7 * (d[p.X\u2080\u2082 # X'\u2082] - d[p.X\u2080\u2082 # X\u2082])\n      \u2264 d[X'\u2081 # X'\u2082] := by\n  have Z := is_tau_min p h h1 h2\n  simp [tau] at Z\n  linarith\n\n", "theoremStatement": "/-- Version of `distance_ge_of_min` with the measures made explicit. -/\nlemma distance_ge_of_min' {\u03a9'\u2081 \u03a9'\u2082 : Type*} (h : tau_minimizes p X\u2081 X\u2082)\n    [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082] {\u03bc : Measure \u03a9'\u2081} {\u03bc' : Measure \u03a9'\u2082}\n    [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] {X'\u2081: \u03a9'\u2081 \u2192 G} {X'\u2082: \u03a9'\u2082 \u2192 G}\n    (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n    d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081; \u2119 # X'\u2081; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n      - p.\u03b7 * (d[p.X\u2080\u2082; \u2119 # X'\u2082; \u03bc'] - d[p.X\u2080\u2082 # X\u2082]) \u2264 d[X'\u2081; \u03bc # X'\u2082; \u03bc'] ", "theoremName": "distance_ge_of_min'", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": {"lineInFile": 179, "tokenPositionInFile": 8618, 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"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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  set M1 : MeasureSpace \u03a9'\u2081 := { volume := \u03bc }\n  set M2 : MeasureSpace \u03a9'\u2082 := { volume := \u03bc' }\n  exact distance_ge_of_min p h h1 h2", "proofType": "tactic", "proofLengthLines": 3, "proofLengthTokens": 137}}
{"srcContext": "import PFR.ForMathlib.CompactProb\nimport PFR.ForMathlib.Entropy.RuzsaDist\n\n/-!\n# The tau functional\n\nDefinition of the tau functional and basic facts\n\n## Main definitions:\n\n* `\u03b7`: $1/9$\n* `\u03c4`: The tau functional $\\tau[X_1; X_2] = d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$\n\n## Main results\n\n* `tau_minimizer_exists`: A pair of random variables minimizing $\\tau$ exists.\n* `condRuzsaDistance_ge_of_min`: If $X_1,X_2$ is a tau-minimizer with $k = d[X_1;X_2]$, then $d[X'_1|Z, X'_2|W]$ is at least\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] )$$\nfor any $X'_1, Z, X'_2, W$.\n-/\n\nopen MeasureTheory ProbabilityTheory\nuniverse uG\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)]\nvariable (G : Type uG) [AddCommGroup G] [ElementaryAddCommGroup G 2] [Fintype G] [MeasurableSpace G]\n\n/-- A structure that packages all the fixed information in the main argument. In this way, when\ndefining the \u03c4 functional, we will only only need to refer to the package once in the notation\ninstead of stating the reference spaces, the reference measures and the reference random\nvariables.\n\nThe \u03b7 parameter has now been incorporated into the package, in preparation for being able to\nmanipulate the package. -/\nstructure refPackage :=\n  /-- The first variable in a package. -/\n  X\u2080\u2081 : \u03a9\u2080\u2081 \u2192 G\n  /-- The second variable in a package. -/\n  X\u2080\u2082 : \u03a9\u2080\u2082 \u2192 G\n  hmeas1 : Measurable X\u2080\u2081\n  hmeas2 : Measurable X\u2080\u2082\n  \u03b7 : \u211d\n  h\u03b7 : 0 < \u03b7\n  h\u03b7' : 8 * \u03b7 \u2264 1\n\nvariable (p : refPackage \u03a9\u2080\u2081 \u03a9\u2080\u2082 G)\nvariable {\u03a9\u2080\u2081 \u03a9\u2080\u2082 G}\n\nvariable {\u03a9\u2081 \u03a9\u2082 \u03a9'\u2081 \u03a9'\u2082 : Type*}\n\n\n/-- If $X_1,X_2$ are two $G$-valued random variables, then\n$$ \\tau[X_1; X_2] := d[X_1; X_2] + \\eta d[X^0_1; X_1] + \\eta d[X^0_2; X_2].$$\nHere, $X^0_1$ and $X^0_2$ are two random variables fixed once and for all in most of the argument.\nTo lighten notation, We package `X^0_1` and `X^0_2` in a single object named `p`.\n\nWe denote it as `\u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p]` where `p` is a fixed package containing the information\nof the reference random variables. When the measurable spaces have a canonical measure `\u2119`, we\ncan use `\u03c4[X\u2081 # X\u2082 | p]`\n-/\n@[pp_dot] noncomputable def tau {\u03a9\u2081 \u03a9\u2082 : Type*} [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    (X\u2081 : \u03a9\u2081 \u2192 G) (X\u2082 : \u03a9\u2082 \u2192 G) (\u03bc\u2081 : Measure \u03a9\u2081) (\u03bc\u2082 : Measure \u03a9\u2082) : \u211d :=\n  d[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082] + p.\u03b7 * d[p.X\u2080\u2081 ; \u2119 # X\u2081 ; \u03bc\u2081] + p.\u03b7 * d[p.X\u2080\u2082 ; \u2119 # X\u2082 ; \u03bc\u2082]\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" ; \" \u03bc\u2081 \" # \" X\u2082 \" ; \" \u03bc\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 \u03bc\u2081 \u03bc\u2082\n\n@[inherit_doc tau]\nnotation3:max \"\u03c4[\" X\u2081 \" # \" X\u2082 \" | \" p\"]\" => tau p X\u2081 X\u2082 MeasureTheory.MeasureSpace.volume MeasureTheory.MeasureSpace.volume\n\nlemma continuous_tau_restrict_probabilityMeasure\n    [TopologicalSpace G] [DiscreteTopology G] [BorelSpace G] :\n    Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p]) := by\n  have obs\u2081 : 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  have obs\u2082 : Continuous\n      (fun (\u03bc : ProbabilityMeasure G \u00d7 ProbabilityMeasure G) \u21a6 d[id ; \u03bc.1.toMeasure # id ; \u03bc.2]) :=\n    continuous_rdist_restrict_probabilityMeasure\n  have obs\u2083 : 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  continuity\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $\\tau[X'_1;X'_2] = \\tau[X_1;X_2]$. -/\nlemma ProbabilityTheory.IdentDistrib.tau_eq [MeasurableSpace \u03a9\u2081] [MeasurableSpace \u03a9\u2082]\n    [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082]\n    {\u03bc\u2081 : Measure \u03a9\u2081} {\u03bc\u2082 : Measure \u03a9\u2082} {\u03bc'\u2081 : Measure \u03a9'\u2081} {\u03bc'\u2082 : Measure \u03a9'\u2082}\n    {X\u2081 : \u03a9\u2081 \u2192 G} {X\u2082 : \u03a9\u2082 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X'\u2081 \u03bc\u2081 \u03bc'\u2081) (h\u2082 : IdentDistrib X\u2082 X'\u2082 \u03bc\u2082 \u03bc'\u2082) :\n    \u03c4[X\u2081 ; \u03bc\u2081 # X\u2082 ; \u03bc\u2082 | p] = \u03c4[X'\u2081 ; \u03bc'\u2081 # X'\u2082 ; \u03bc'\u2082 | p] := by\n  simp only [tau]\n  rw [(IdentDistrib.refl p.hmeas1.aemeasurable).rdist_eq h\u2081,\n      (IdentDistrib.refl p.hmeas2.aemeasurable).rdist_eq h\u2082,\n      h\u2081.rdist_eq h\u2082]\n\n/-- Property recording the fact that two random variables minimize the tau functional. Expressed\nin terms of measures on the group to avoid quantifying over all spaces, but this implies comparison\nwith any pair of random variables, see Lemma `is_tau_min`. -/\ndef tau_minimizes {\u03a9 : Type*} [MeasureSpace \u03a9] (X\u2081 : \u03a9 \u2192 G) (X\u2082 : \u03a9 \u2192 G) : Prop :=\n  \u2200 (\u03bd\u2081 : Measure G) (\u03bd\u2082 : Measure G), IsProbabilityMeasure \u03bd\u2081 \u2192 IsProbabilityMeasure \u03bd\u2082 \u2192\n      \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p]\n\n/-- If $X'_1, X'_2$ are copies of $X_1,X_2$, then $X_1, X_2$ minimize $\\tau$ iff $X_1', X_2'$ do. -/\nlemma ProbabilityTheory.IdentDistrib.tau_minimizes [MeasureSpace \u03a9]\n    [MeasureSpace \u03a9']\n    {X\u2081 X\u2082 : \u03a9 \u2192 G} {X\u2081' X\u2082' : \u03a9' \u2192 G}\n    (h\u2081 : IdentDistrib X\u2081 X\u2081') (h\u2082 : IdentDistrib X\u2082 X\u2082') :\n    tau_minimizes p X\u2081 X\u2082 \u2194 tau_minimizes p X\u2081' X\u2082' := by\n  simp_rw [_root_.tau_minimizes, h\u2081.tau_eq p h\u2082]\n\n/-- A pair of measures minimizing $\\tau$ exists. -/\nlemma tau_min_exists_measure [MeasurableSingletonClass G] :\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 | p] \u2264 \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] := by\n  let _i : TopologicalSpace G := (\u22a5 : TopologicalSpace G) -- Equip G with the discrete topology.\n  have : DiscreteTopology G := \u27e8rfl\u27e9\n  have GG_cpt : CompactSpace (ProbabilityMeasure G \u00d7 ProbabilityMeasure G) := inferInstance\n  let T : ProbabilityMeasure G \u00d7 ProbabilityMeasure G \u2192 \u211d := -- restrict \u03c4 to the compact subspace\n    fun \u27e8\u03bc\u2081, \u03bc\u2082\u27e9 \u21a6 \u03c4[id ; \u03bc\u2081 # id ; \u03bc\u2082 | p]\n  have T_cont : Continuous T := by apply continuous_tau_restrict_probabilityMeasure\n  haveI : Inhabited G := \u27e80\u27e9 -- Need to record this for Lean to know that proba measures exist.\n  obtain \u27e8\u03bc, _, h\u03bc\u27e9 := @IsCompact.exists_isMinOn \u211d (ProbabilityMeasure G \u00d7 ProbabilityMeasure G)\n                          _ _ _ _ Set.univ isCompact_univ \u27e8default, trivial\u27e9 T T_cont.continuousOn\n  use \u27e8\u03bc.1.toMeasure, \u03bc.2.toMeasure\u27e9\n  refine \u27e8\u03bc.1.prop, \u03bc.2.prop, ?_\u27e9\n  intro \u03bd\u2081 \u03bd\u2082 P\u03bd\u2081 P\u03bd\u2082\n  rw [isMinOn_univ_iff] at h\u03bc\n  let \u03bd : ProbabilityMeasure G \u00d7 ProbabilityMeasure G := \u27e8\u27e8\u03bd\u2081, P\u03bd\u2081\u27e9, \u03bd\u2082, P\u03bd\u2082\u27e9\n  exact h\u03bc \u03bd\n\n/-- A pair of random variables minimizing $\u03c4$ exists. -/\nlemma tau_minimizer_exists [MeasurableSingletonClass G] :\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\n    tau_minimizes p X\u2081 X\u2082 := by\n  let \u03bc := (tau_min_exists_measure p).choose\n  have : IsProbabilityMeasure \u03bc.1 := (tau_min_exists_measure p).choose_spec.1\n  have : IsProbabilityMeasure \u03bc.2 := (tau_min_exists_measure p).choose_spec.2.1\n  have P : IsProbabilityMeasure (\u03bc.1.prod \u03bc.2) := by infer_instance\n  let M : MeasureSpace (G \u00d7 G) := \u27e8\u03bc.1.prod \u03bc.2\u27e9\n  refine \u27e8G \u00d7 G, M, Prod.fst, Prod.snd, measurable_fst, measurable_snd, P, ?_\u27e9\n  intro \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n  have A : \u03c4[@Prod.fst G G # @Prod.snd G G | p] = \u03c4[id ; \u03bc.1 # id ; \u03bc.2 | p] :=\n    ProbabilityTheory.IdentDistrib.tau_eq p IdentDistrib.fst_id IdentDistrib.snd_id\n  convert (tau_min_exists_measure p).choose_spec.2.2 \u03bd\u2081 \u03bd\u2082 h\u2081 h\u2082\n\n\nvariable [MeasureSpace \u03a9] [h\u03a9\u2081 : MeasureSpace \u03a9'\u2081] [h\u03a9\u2082 : MeasureSpace \u03a9'\u2082]\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9)]\n  [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2081)] [IsProbabilityMeasure (\u2119 : Measure \u03a9'\u2082)]\n  {X\u2081 : \u03a9 \u2192 G} {X\u2082 : \u03a9 \u2192 G} {X'\u2081 : \u03a9'\u2081 \u2192 G} {X'\u2082 : \u03a9'\u2082 \u2192 G}\n\nlemma is_tau_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n    \u03c4[X\u2081 # X\u2082 | p] \u2264 \u03c4[X'\u2081 # X'\u2082 | p] := by\n  let \u03bd\u2081 := (\u2119 : Measure \u03a9'\u2081).map X'\u2081\n  let \u03bd\u2082 := (\u2119 : Measure \u03a9'\u2082).map X'\u2082\n  have B : \u03c4[X'\u2081 # X'\u2082 | p] = \u03c4[id ; \u03bd\u2081 # id ; \u03bd\u2082 | p] :=\n    (identDistrib_id_right h1.aemeasurable).tau_eq p (identDistrib_id_right h2.aemeasurable)\n  convert h \u03bd\u2081 \u03bd\u2082 (isProbabilityMeasure_map h1.aemeasurable)\n    (isProbabilityMeasure_map h2.aemeasurable)\n\n/-- Let `X\u2081` and `X\u2082` be tau-minimizers associated to `p`, with $d[X_1,X_2]=k$, then\n$$ d[X'_1;X'_2] \\geq\n    k - \\eta (d[X^0_1;X'_1] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2] - d[X^0_2;X_2] )$$\nfor any $G$-valued random variables $X'_1,X'_2$.\n-/\nlemma distance_ge_of_min (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n    d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081 # X'\u2081] - d[p.X\u2080\u2081 # X\u2081]) - p.\u03b7 * (d[p.X\u2080\u2082 # X'\u2082] - d[p.X\u2080\u2082 # X\u2082])\n      \u2264 d[X'\u2081 # X'\u2082] := by\n  have Z := is_tau_min p h h1 h2\n  simp [tau] at Z\n  linarith\n\n/-- Version of `distance_ge_of_min` with the measures made explicit. -/\nlemma distance_ge_of_min' {\u03a9'\u2081 \u03a9'\u2082 : Type*} (h : tau_minimizes p X\u2081 X\u2082)\n    [MeasurableSpace \u03a9'\u2081] [MeasurableSpace \u03a9'\u2082] {\u03bc : Measure \u03a9'\u2081} {\u03bc' : Measure \u03a9'\u2082}\n    [IsProbabilityMeasure \u03bc] [IsProbabilityMeasure \u03bc'] {X'\u2081: \u03a9'\u2081 \u2192 G} {X'\u2082: \u03a9'\u2082 \u2192 G}\n    (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082) :\n    d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081; \u2119 # X'\u2081; \u03bc] - d[p.X\u2080\u2081 # X\u2081])\n      - p.\u03b7 * (d[p.X\u2080\u2082; \u2119 # X'\u2082; \u03bc'] - d[p.X\u2080\u2082 # X\u2082]) \u2264 d[X'\u2081; \u03bc # X'\u2082; \u03bc'] := by\n  set M1 : MeasureSpace \u03a9'\u2081 := { volume := \u03bc }\n  set M2 : MeasureSpace \u03a9'\u2082 := { volume := \u03bc' }\n  exact distance_ge_of_min p h h1 h2\n\n\nopen BigOperators\n\n", "theoremStatement": "/-- For any $G$-valued random variables $X'_1,X'_2$ and random variables $Z,W$, one can lower\nbound $d[X'_1|Z;X'_2|W]$ by\n$$k - \\eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \\eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] ).$$\n-/\nlemma condRuzsaDistance_ge_of_min [MeasurableSingletonClass G]\n    [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S]\n    [Fintype T] [MeasurableSpace T] [MeasurableSingletonClass T]\n    (h : tau_minimizes p X\u2081 X\u2082) (h1 : Measurable X'\u2081) (h2 : Measurable X'\u2082)\n    (Z : \u03a9'\u2081 \u2192 S) (W : \u03a9'\u2082 \u2192 T) (hZ : Measurable Z) (hW : Measurable W) :\n    d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081 # X'\u2081 | Z] - d[p.X\u2080\u2081 # X\u2081])\n      - p.\u03b7 * (d[p.X\u2080\u2082 # X'\u2082 | W] - d[p.X\u2080\u2082 # X\u2082]) \u2264 d[X'\u2081 | Z # X'\u2082 | W] ", "theoremName": "condRuzsaDistance_ge_of_min", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "12546ab", "date": "2023-12-20"}, "file": "PFR/PFR/TauFunctional.lean", "module": "PFR.TauFunctional", "jsonFile": "PFR.TauFunctional.jsonl", "positionMetadata": 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"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", "PFR.ForMathlib.MeasureReal", "Mathlib.Topology.Algebra.Module.WeakDual", "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.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", "PFR.Mathlib.MeasureTheory.Measure.ProbabilityMeasure", "PFR.ForMathlib.FiniteMeasureComponent", "PFR.ForMathlib.CompactProb", "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.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", "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.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", "PFR.ForMathlib.FiniteMeasureProd", "PFR.ForMathlib.ProbabilityMeasureProdCont", "PFR.Mathlib.Data.Fin.VecNotation", "PFR.ForMathlib.Entropy.RuzsaDist"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have hz (a : \u211d) : a = \u2211 z in FiniteRange.toFinset Z, (\u2119 (Z \u207b\u00b9' {z})).toReal * a := by\n    simp_rw [\u2190 Finset.sum_mul,\u2190 Measure.map_apply hZ (MeasurableSet.singleton _), Finset.sum_toReal_measure_singleton]\n    rw [FiniteRange.full hZ]\n    simp\n  have hw (a : \u211d) : a = \u2211 w in FiniteRange.toFinset W, (\u2119 (W \u207b\u00b9' {w})).toReal * a := by\n    simp_rw [\u2190 Finset.sum_mul,\u2190 Measure.map_apply hW (MeasurableSet.singleton _), Finset.sum_toReal_measure_singleton]\n    rw [FiniteRange.full hW]\n    simp\n  rw [condRuzsaDist_eq_sum h1 hZ h2 hW, condRuzsaDist'_eq_sum h1 hZ, hz d[X\u2081 # X\u2082],\n    hz d[p.X\u2080\u2081 # X\u2081], hz (p.\u03b7 * (d[p.X\u2080\u2082 # X'\u2082 | W] - d[p.X\u2080\u2082 # X\u2082])),\n    \u2190 Finset.sum_sub_distrib, Finset.mul_sum, \u2190 Finset.sum_sub_distrib, \u2190 Finset.sum_sub_distrib]\n  apply Finset.sum_le_sum\n  intro z _\n  rw [condRuzsaDist'_eq_sum h2 hW, hw d[p.X\u2080\u2082 # X\u2082],\n    hw ((\u2119 (Z \u207b\u00b9' {z})).toReal * d[X\u2081 # X\u2082] - p.\u03b7 * ((\u2119 (Z \u207b\u00b9' {z})).toReal *\n      d[p.X\u2080\u2081 ; \u2119 # X'\u2081 ; \u2119[|Z \u2190 z]] - (\u2119 (Z \u207b\u00b9' {z})).toReal * d[p.X\u2080\u2081 # X\u2081])),\n    \u2190 Finset.sum_sub_distrib, Finset.mul_sum, Finset.mul_sum, \u2190 Finset.sum_sub_distrib]\n  apply Finset.sum_le_sum\n  intro w _\n  rcases eq_or_ne (\u2119 (Z \u207b\u00b9' {z})) 0 with hpz | hpz\n  . simp [hpz]\n  rcases eq_or_ne (\u2119 (W \u207b\u00b9' {w})) 0 with hpw | hpw\n  . simp [hpw]\n  set \u03bc := (h\u03a9\u2081.volume)[|Z \u2190 z]\n  have h\u03bc : IsProbabilityMeasure \u03bc := cond_isProbabilityMeasure \u2119 hpz\n  set \u03bc' := \u2119[|W \u2190 w]\n  have h\u03bc' : IsProbabilityMeasure \u03bc' := cond_isProbabilityMeasure \u2119 hpw\n  suffices d[X\u2081 # X\u2082] - p.\u03b7 * (d[p.X\u2080\u2081; volume # X'\u2081; \u03bc] - d[p.X\u2080\u2081 # X\u2081]) -\n    p.\u03b7 * (d[p.X\u2080\u2082; volume # X'\u2082; \u03bc'] - d[p.X\u2080\u2082 # X\u2082]) \u2264 d[X'\u2081 ; \u03bc # X'\u2082; \u03bc'] by\n    replace this := mul_le_mul_of_nonneg_left this (show 0 \u2264 (\u2119 (Z \u207b\u00b9' {z})).toReal * (\u2119 (W \u207b\u00b9' {w})).toReal by positivity)\n    convert this using 1\n    ring\n  exact distance_ge_of_min' p h h1 h2", "proofType": "tactic", "proofLengthLines": 33, "proofLengthTokens": 1805}}
{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\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\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n    [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n    [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n      = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n  have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n    ext \u03c9; simp [sub_eq_zero, eq_comm]\n  rw [this, measureReal_iUnion_fintype _\n    (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n  \u00b7 intro g g' hgg'\n    apply Set.disjoint_iff_inter_eq_empty.2\n    ext a\n    simp (config := {contextual := True}) [hgg']\n  classical\n  let W : Finset G := A \u2229 B\n  calc\n    \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n      = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply sum_congr _ _ (fun g \u21a6 ?_)\n        rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n          measurableSet_discrete _]\n    _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply (Finset.sum_subset W.subset_univ _).symm\n        intro i _ hi\n        replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n        rcases hi with h'i|h'i\n        \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n        \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n    _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n        apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n        replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n        rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n            Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n    _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n    _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n        congr\n        rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n        simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n      = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n  classical\n  let W := fun \u03c9 \u21a6 V \u03c9 + x\n  have Wunif : IsUniform (B + {x} : Set G) W := by\n    convert Vunif.comp (add_left_injective x)\n    simp\n  have Wmeas : Measurable W := Vmeas.add_const _\n  have UWindep : IndepFun U W := by\n    have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n    exact hindep.comp measurable_id this\n  have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n    ext \u03c9\n    simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n    abel_nf\n  have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n  rw [h] at Wunif\n  rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n  congr 3\n  . rw [add_singleton]; simp\n  convert Finset.card_vadd_finset (AddOpposite.op x) B\n  . simp\n  simp\n\nend ProbabilityTheory\n\n\n", "theoremStatement": "/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K ", "theoremName": "PFR_conjecture_pos_aux", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "2ac9116", "date": "2023-12-26"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 100, "tokenPositionInFile": 4248, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": false, "numRepositoryPremises": 0, "numPremises": 98, "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", 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"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"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n    have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n    have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 522}}
{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\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\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n    [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n    [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n      = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n  have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n    ext \u03c9; simp [sub_eq_zero, eq_comm]\n  rw [this, measureReal_iUnion_fintype _\n    (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n  \u00b7 intro g g' hgg'\n    apply Set.disjoint_iff_inter_eq_empty.2\n    ext a\n    simp (config := {contextual := True}) [hgg']\n  classical\n  let W : Finset G := A \u2229 B\n  calc\n    \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n      = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply sum_congr _ _ (fun g \u21a6 ?_)\n        rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n          measurableSet_discrete _]\n    _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply (Finset.sum_subset W.subset_univ _).symm\n        intro i _ hi\n        replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n        rcases hi with h'i|h'i\n        \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n        \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n    _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n        apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n        replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n        rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n            Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n    _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n    _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n        congr\n        rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n        simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n      = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n  classical\n  let W := fun \u03c9 \u21a6 V \u03c9 + x\n  have Wunif : IsUniform (B + {x} : Set G) W := by\n    convert Vunif.comp (add_left_injective x)\n    simp\n  have Wmeas : Measurable W := Vmeas.add_const _\n  have UWindep : IndepFun U W := by\n    have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n    exact hindep.comp measurable_id this\n  have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n    ext \u03c9\n    simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n    abel_nf\n  have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n  rw [h] at Wunif\n  rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n  congr 3\n  . rw [add_singleton]; simp\n  convert Finset.card_vadd_finset (AddOpposite.op x) B\n  . simp\n  simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n    have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n    have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\n", "theoremStatement": "lemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K ", "theoremName": 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"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"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n    have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n    have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9", "proofType": "tactic", "proofLengthLines": 8, "proofLengthTokens": 522}}
{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\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\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n    [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n    [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n      = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n  have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n    ext \u03c9; simp [sub_eq_zero, eq_comm]\n  rw [this, measureReal_iUnion_fintype _\n    (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n  \u00b7 intro g g' hgg'\n    apply Set.disjoint_iff_inter_eq_empty.2\n    ext a\n    simp (config := {contextual := True}) [hgg']\n  classical\n  let W : Finset G := A \u2229 B\n  calc\n    \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n      = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply sum_congr _ _ (fun g \u21a6 ?_)\n        rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n          measurableSet_discrete _]\n    _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply (Finset.sum_subset W.subset_univ _).symm\n        intro i _ hi\n        replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n        rcases hi with h'i|h'i\n        \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n        \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n    _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n        apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n        replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n        rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n            Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n    _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n    _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n        congr\n        rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n        simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n      = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n  classical\n  let W := fun \u03c9 \u21a6 V \u03c9 + x\n  have Wunif : IsUniform (B + {x} : Set G) W := by\n    convert Vunif.comp (add_left_injective x)\n    simp\n  have Wmeas : Measurable W := Vmeas.add_const _\n  have UWindep : IndepFun U W := by\n    have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n    exact hindep.comp measurable_id this\n  have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n    ext \u03c9\n    simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n    abel_nf\n  have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n  rw [h] at Wunif\n  rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n  congr 3\n  . rw [add_singleton]; simp\n  convert Finset.card_vadd_finset (AddOpposite.op x) B\n  . simp\n  simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n    have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n    have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nlemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K := by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n    have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n    have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nvariable {G : Type*} [AddCommGroup G] [MeasurableSpace G]\n  [MeasurableSingletonClass G] {A : Set G} [Finite A] {K : \u211d} [Countable G]\n\n", "theoremStatement": "/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A)\n    {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U\u2080 : \u03a9 \u2192 G}\n    (U\u2080unif : IsUniform A U\u2080) (U\u2080meas : Measurable U\u2080) : d[U\u2080 # U\u2080] \u2264 log K ", "theoremName": "rdist_le_of_isUniform_of_card_add_le", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "2ac9116", "date": "2023-12-26"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 126, "tokenPositionInFile": 5928, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 1, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 263, "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", 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"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"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  obtain \u27e8A_pos, AA_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  rcases independent_copies_two U\u2080meas U\u2080meas with \u27e8\u03a9, m\u03a9, U, U', hP, hU, hU', UU'_indep, idU, idU'\u27e9\n  have Uunif : IsUniform A U := U\u2080unif.of_identDistrib idU.symm $ measurableSet_discrete _\n  have U'unif : IsUniform A U' := U\u2080unif.of_identDistrib idU'.symm $ measurableSet_discrete _\n  have IU : d[U # U'] \u2264 log K := by\n    have I : H[U - U'] \u2264 log (Nat.card (A - A)) := by\n      convert entropy_le_log_card_of_mem (A := (A-A).toFinite.toFinset) ?_ ?_ with x\n      . simp\n        exact Iff.rfl\n      . measurability\n      filter_upwards [Uunif.ae_mem, U'unif.ae_mem] with \u03c9 h1 h2\n      simp\n      exact Set.sub_mem_sub h1 h2\n    have J : log (Nat.card (A - A)) \u2264 log K + log (Nat.card A) := by\n      apply (log_le_log AA_pos hA).trans (le_of_eq _)\n      rw [log_mul K_pos.ne' A_pos.ne']\n--    have : H[U + U'] = H[U - U'] := by congr; simp\n    rw [UU'_indep.rdist_eq hU hU', IsUniform.entropy_eq' Uunif hU, IsUniform.entropy_eq' U'unif hU']\n    linarith\n  rwa [idU.rdist_eq idU'] at IU", "proofType": "tactic", "proofLengthLines": 21, "proofLengthTokens": 1122}}
{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\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\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n    [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n    [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n      = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n  have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n    ext \u03c9; simp [sub_eq_zero, eq_comm]\n  rw [this, measureReal_iUnion_fintype _\n    (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n  \u00b7 intro g g' hgg'\n    apply Set.disjoint_iff_inter_eq_empty.2\n    ext a\n    simp (config := {contextual := True}) [hgg']\n  classical\n  let W : Finset G := A \u2229 B\n  calc\n    \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n      = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply sum_congr _ _ (fun g \u21a6 ?_)\n        rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n          measurableSet_discrete _]\n    _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply (Finset.sum_subset W.subset_univ _).symm\n        intro i _ hi\n        replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n        rcases hi with h'i|h'i\n        \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n        \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n    _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n        apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n        replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n        rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n            Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n    _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n    _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n        congr\n        rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n        simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n      = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n  classical\n  let W := fun \u03c9 \u21a6 V \u03c9 + x\n  have Wunif : IsUniform (B + {x} : Set G) W := by\n    convert Vunif.comp (add_left_injective x)\n    simp\n  have Wmeas : Measurable W := Vmeas.add_const _\n  have UWindep : IndepFun U W := by\n    have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n    exact hindep.comp measurable_id this\n  have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n    ext \u03c9\n    simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n    abel_nf\n  have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n  rw [h] at Wunif\n  rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n  congr 3\n  . rw [add_singleton]; simp\n  convert Finset.card_vadd_finset (AddOpposite.op x) B\n  . simp\n  simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n    have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n    have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nlemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K := by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n    have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n    have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nvariable {G : Type*} [AddCommGroup G] [MeasurableSpace G]\n  [MeasurableSingletonClass G] {A : Set G} [Finite A] {K : \u211d} [Countable G]\n\n/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A)\n    {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U\u2080 : \u03a9 \u2192 G}\n    (U\u2080unif : IsUniform A U\u2080) (U\u2080meas : Measurable U\u2080) : d[U\u2080 # U\u2080] \u2264 log K := by\n  obtain \u27e8A_pos, AA_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  rcases independent_copies_two U\u2080meas U\u2080meas with \u27e8\u03a9, m\u03a9, U, U', hP, hU, hU', UU'_indep, idU, idU'\u27e9\n  have Uunif : IsUniform A U := U\u2080unif.of_identDistrib idU.symm $ measurableSet_discrete _\n  have U'unif : IsUniform A U' := U\u2080unif.of_identDistrib idU'.symm $ measurableSet_discrete _\n  have IU : d[U # U'] \u2264 log K := by\n    have I : H[U - U'] \u2264 log (Nat.card (A - A)) := by\n      convert entropy_le_log_card_of_mem (A := (A-A).toFinite.toFinset) ?_ ?_ with x\n      . simp\n        exact Iff.rfl\n      . measurability\n      filter_upwards [Uunif.ae_mem, U'unif.ae_mem] with \u03c9 h1 h2\n      simp\n      exact Set.sub_mem_sub h1 h2\n    have J : log (Nat.card (A - A)) \u2264 log K + log (Nat.card A) := by\n      apply (log_le_log AA_pos hA).trans (le_of_eq _)\n      rw [log_mul K_pos.ne' A_pos.ne']\n--    have : H[U + U'] = H[U - U'] := by congr; simp\n    rw [UU'_indep.rdist_eq hU hU', IsUniform.entropy_eq' Uunif hU, IsUniform.entropy_eq' U'unif hU']\n    linarith\n  rwa [idU.rdist_eq idU'] at IU\n\nvariable [ElementaryAddCommGroup G 2] [Fintype G]\n\n", "theoremStatement": "lemma sumset_eq_sub : A + A = A - A ", "theoremName": "sumset_eq_sub", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "93e5ee0", "date": "2024-01-12"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 155, "tokenPositionInFile": 7476, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "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", 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"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"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  rw [\u2190 Set.image2_add, \u2190 Set.image2_sub]\n  congr! 1 with a _ b _\n  show a + b = a - b\n  simp", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 99}}
{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\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\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n    [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n    [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n      = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n  have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n    ext \u03c9; simp [sub_eq_zero, eq_comm]\n  rw [this, measureReal_iUnion_fintype _\n    (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n  \u00b7 intro g g' hgg'\n    apply Set.disjoint_iff_inter_eq_empty.2\n    ext a\n    simp (config := {contextual := True}) [hgg']\n  classical\n  let W : Finset G := A \u2229 B\n  calc\n    \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n      = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply sum_congr _ _ (fun g \u21a6 ?_)\n        rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n          measurableSet_discrete _]\n    _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply (Finset.sum_subset W.subset_univ _).symm\n        intro i _ hi\n        replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n        rcases hi with h'i|h'i\n        \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n        \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n    _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n        apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n        replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n        rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n            Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n    _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n    _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n        congr\n        rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n        simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n      = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n  classical\n  let W := fun \u03c9 \u21a6 V \u03c9 + x\n  have Wunif : IsUniform (B + {x} : Set G) W := by\n    convert Vunif.comp (add_left_injective x)\n    simp\n  have Wmeas : Measurable W := Vmeas.add_const _\n  have UWindep : IndepFun U W := by\n    have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n    exact hindep.comp measurable_id this\n  have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n    ext \u03c9\n    simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n    abel_nf\n  have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n  rw [h] at Wunif\n  rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n  congr 3\n  . rw [add_singleton]; simp\n  convert Finset.card_vadd_finset (AddOpposite.op x) B\n  . simp\n  simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n    have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n    have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nlemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K := by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n    have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n    have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nvariable {G : Type*} [AddCommGroup G] [MeasurableSpace G]\n  [MeasurableSingletonClass G] {A : Set G} [Finite A] {K : \u211d} [Countable G]\n\n/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A)\n    {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U\u2080 : \u03a9 \u2192 G}\n    (U\u2080unif : IsUniform A U\u2080) (U\u2080meas : Measurable U\u2080) : d[U\u2080 # U\u2080] \u2264 log K := by\n  obtain \u27e8A_pos, AA_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  rcases independent_copies_two U\u2080meas U\u2080meas with \u27e8\u03a9, m\u03a9, U, U', hP, hU, hU', UU'_indep, idU, idU'\u27e9\n  have Uunif : IsUniform A U := U\u2080unif.of_identDistrib idU.symm $ measurableSet_discrete _\n  have U'unif : IsUniform A U' := U\u2080unif.of_identDistrib idU'.symm $ measurableSet_discrete _\n  have IU : d[U # U'] \u2264 log K := by\n    have I : H[U - U'] \u2264 log (Nat.card (A - A)) := by\n      convert entropy_le_log_card_of_mem (A := (A-A).toFinite.toFinset) ?_ ?_ with x\n      . simp\n        exact Iff.rfl\n      . measurability\n      filter_upwards [Uunif.ae_mem, U'unif.ae_mem] with \u03c9 h1 h2\n      simp\n      exact Set.sub_mem_sub h1 h2\n    have J : log (Nat.card (A - A)) \u2264 log K + log (Nat.card A) := by\n      apply (log_le_log AA_pos hA).trans (le_of_eq _)\n      rw [log_mul K_pos.ne' A_pos.ne']\n--    have : H[U + U'] = H[U - U'] := by congr; simp\n    rw [UU'_indep.rdist_eq hU hU', IsUniform.entropy_eq' Uunif hU, IsUniform.entropy_eq' U'unif hU']\n    linarith\n  rwa [idU.rdist_eq idU'] at IU\n\nvariable [ElementaryAddCommGroup G 2] [Fintype G]\n\nlemma sumset_eq_sub : A + A = A - A := by\n  rw [\u2190 Set.image2_add, \u2190 Set.image2_sub]\n  congr! 1 with a _ b _\n  show a + b = a - b\n  simp\n\n", "theoremStatement": "/-- 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 $K^{13/2} |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^11$. -/\nlemma PFR_conjecture_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 ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n      \u2227 Nat.card H \u2264 K ^ 11 * Nat.card A \u2227 Nat.card A \u2264 K ^ 11 * Nat.card H \u2227 A \u2286 c + H ", "theoremName": "PFR_conjecture_aux", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "b94162e", "date": "2023-11-28"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 161, "tokenPositionInFile": 7613, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 33, "numPremises": 483, "importedModules": ["Init.Prelude", "Init.Coe", "Init.Notation", "Init.Tactics", "Init.SizeOf", "Init.Core", "Init.MetaTypes", "Init.SimpLemmas", "Init.Data.Nat.Basic", "Init.WF", 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"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"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  classical\n  let _mG : MeasurableSpace G := \u22a4\n  rw [sumset_eq_sub] at hA\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 : d[UA # UA] \u2264 log K := rdist_le_of_isUniform_of_card_add_le h\u2080A hA UAunif UAmeas\n  rw [\u2190 sumset_eq_sub] at hA\n  let p : refPackage \u03a9\u2080 \u03a9\u2080 G := \u27e8UA, UA, UAmeas, UAmeas, 1/9, (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 p (by norm_num) 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 11/2 * 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 11 * 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 ^ 11 * Nat.card (H : Set G) := by\n    have : log (Nat.card A) \u2264 log K * 11 + 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 ^ 11 * Nat.card A := by\n    have : log (Nat.card (H : Set G)) \u2264 log K * 11 + 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 * (-11/2) + 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 ^ (-11/2) * (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 ^ (13/2) * (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 ^ (11/2) * 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    rw [add_sub_assoc] at Au\n    refine Au.trans $ add_subset_add_left $\n      (sub_subset_sub (inter_subset_right ..) (inter_subset_right ..)).trans ?_\n    rw [add_sub_add_comm, singleton_sub_singleton, sub_self]\n    simp\n  exact \u27e8H, u, Iu, IHA, IAH, A_subset_uH\u27e9", "proofType": "tactic", "proofLengthLines": 107, "proofLengthTokens": 5872}}
{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\n\n/- In this file the power notation will always mean the base and exponent are real numbers. -/\nlocal macro_rules | `($x ^ $y) => `(HPow.hPow ($x : \u211d) ($y : \u211d))\n\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n    [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n    [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n      = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n  have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n    ext \u03c9; simp [sub_eq_zero, eq_comm]\n  rw [this, measureReal_iUnion_fintype _\n    (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n  \u00b7 intro g g' hgg'\n    apply Set.disjoint_iff_inter_eq_empty.2\n    ext a\n    simp (config := {contextual := True}) [hgg']\n  classical\n  let W : Finset G := A \u2229 B\n  calc\n    \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n      = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply sum_congr _ _ (fun g \u21a6 ?_)\n        rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n          measurableSet_discrete _]\n    _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply (Finset.sum_subset W.subset_univ _).symm\n        intro i _ hi\n        replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n        rcases hi with h'i|h'i\n        \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n        \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n    _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n        apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n        replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n        rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n            Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n    _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n    _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n        congr\n        rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n        simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n      = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n  classical\n  let W := fun \u03c9 \u21a6 V \u03c9 + x\n  have Wunif : IsUniform (B + {x} : Set G) W := by\n    convert Vunif.comp (add_left_injective x)\n    simp\n  have Wmeas : Measurable W := Vmeas.add_const _\n  have UWindep : IndepFun U W := by\n    have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n    exact hindep.comp measurable_id this\n  have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n    ext \u03c9\n    simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n    abel_nf\n  have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n  rw [h] at Wunif\n  rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n  congr 3\n  . rw [add_singleton]; simp\n  convert Finset.card_vadd_finset (AddOpposite.op x) B\n  . simp\n  simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n    have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n    have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nlemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K := by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n    have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n    have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nvariable {G : Type*} [AddCommGroup G] [MeasurableSpace G]\n  [MeasurableSingletonClass G] {A : Set G} [Finite A] {K : \u211d} [Countable G]\n\n/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A)\n    {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U\u2080 : \u03a9 \u2192 G}\n    (U\u2080unif : IsUniform A U\u2080) (U\u2080meas : Measurable U\u2080) : d[U\u2080 # U\u2080] \u2264 log K := by\n  obtain \u27e8A_pos, AA_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  rcases independent_copies_two U\u2080meas U\u2080meas with \u27e8\u03a9, m\u03a9, U, U', hP, hU, hU', UU'_indep, idU, idU'\u27e9\n  have Uunif : IsUniform A U := U\u2080unif.of_identDistrib idU.symm $ measurableSet_discrete _\n  have U'unif : IsUniform A U' := U\u2080unif.of_identDistrib idU'.symm $ measurableSet_discrete _\n  have IU : d[U # U'] \u2264 log K := by\n    have I : H[U - U'] \u2264 log (Nat.card (A - A)) := by\n      convert entropy_le_log_card_of_mem (A := (A-A).toFinite.toFinset) ?_ ?_ with x\n      . simp\n        exact Iff.rfl\n      . measurability\n      filter_upwards [Uunif.ae_mem, U'unif.ae_mem] with \u03c9 h1 h2\n      simp\n      exact Set.sub_mem_sub h1 h2\n    have J : log (Nat.card (A - A)) \u2264 log K + log (Nat.card A) := by\n      apply (log_le_log AA_pos hA).trans (le_of_eq _)\n      rw [log_mul K_pos.ne' A_pos.ne']\n--    have : H[U + U'] = H[U - U'] := by congr; simp\n    rw [UU'_indep.rdist_eq hU hU', IsUniform.entropy_eq' Uunif hU, IsUniform.entropy_eq' U'unif hU']\n    linarith\n  rwa [idU.rdist_eq idU'] at IU\n\nvariable [ElementaryAddCommGroup G 2] [Fintype G]\n\nlemma sumset_eq_sub : A + A = A - A := by\n  rw [\u2190 Set.image2_add, \u2190 Set.image2_sub]\n  congr! 1 with a _ b _\n  show a + b = a - b\n  simp\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 $K^{13/2} |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^11$. -/\nlemma PFR_conjecture_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 ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n      \u2227 Nat.card H \u2264 K ^ 11 * Nat.card A \u2227 Nat.card A \u2264 K ^ 11 * Nat.card H \u2227 A \u2286 c + H := by\n  classical\n  let _mG : MeasurableSpace G := \u22a4\n  rw [sumset_eq_sub] at hA\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 : d[UA # UA] \u2264 log K := rdist_le_of_isUniform_of_card_add_le h\u2080A hA UAunif UAmeas\n  rw [\u2190 sumset_eq_sub] at hA\n  let p : refPackage \u03a9\u2080 \u03a9\u2080 G := \u27e8UA, UA, UAmeas, UAmeas, 1/9, (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 p (by norm_num) 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 11/2 * 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 11 * 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 ^ 11 * Nat.card (H : Set G) := by\n    have : log (Nat.card A) \u2264 log K * 11 + 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 ^ 11 * Nat.card A := by\n    have : log (Nat.card (H : Set G)) \u2264 log K * 11 + 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 * (-11/2) + 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 ^ (-11/2) * (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 ^ (13/2) * (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 ^ (11/2) * 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    rw [add_sub_assoc] at Au\n    refine Au.trans $ add_subset_add_left $\n      (sub_subset_sub (inter_subset_right ..) (inter_subset_right ..)).trans ?_\n    rw [add_sub_add_comm, singleton_sub_singleton, sub_self]\n    simp\n  exact \u27e8H, u, Iu, IHA, IAH, A_subset_uH\u27e9\n\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 $2K^{12}$ cosets of\na subgroup of cardinality at most $|A|$. -/\ntheorem PFR_conjecture (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 ^ 12 \u2227 Nat.card H \u2264 Nat.card A \u2227 A \u2286 c + H ", "theoremName": "PFR_conjecture", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "b94162e", "date": "2023-11-28"}, "file": "PFR/PFR/Main.lean", "module": "PFR.Main", "jsonFile": "PFR.Main.jsonl", "positionMetadata": {"lineInFile": 278, "tokenPositionInFile": 14154, "theoremPositionInFile": 7}, 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"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"]}, "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 ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n      \u2227 Nat.card (H : Set G) \u2264 K ^ 11 * Nat.card A \u2227 Nat.card A \u2264 K ^ 11 * Nat.card (H : Set G)\n      \u2227 A \u2286 c + H :=\n    PFR_conjecture_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 ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := hc\n    _ \u2264 K ^ (13/2) * (K ^ 11 * Nat.card (H : Set G)) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := by\n      gcongr\n    _ = K ^ 12 := by rpow_ring; norm_num\n    _ < 2 * K ^ 12 := by linarith [show 0 < K ^ 12 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 ^ (13/2) * (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 ^ (13/2) * (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 ^ (13/2) * (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 ^ (13/2) * (Nat.card A) ^ (-1/2) * (K ^ 11 * Nat.card A) ^ (1/2) := by\n        gcongr\n    _ = 2 * K ^ 12 := by\n        rpow_ring\n        norm_num", "proofType": "tactic", "proofLengthLines": 53, "proofLengthTokens": 2786}}
{"srcContext": "import Mathlib.Combinatorics.Additive.RuzsaCovering\nimport Mathlib.GroupTheory.Complement\nimport Mathlib.GroupTheory.OrderOfElement\nimport PFR.Mathlib.GroupTheory.Subgroup.Pointwise\nimport PFR.ForMathlib.Entropy.RuzsaSetDist\nimport PFR.Tactic.RPowSimp\nimport PFR.TauFunctional\nimport PFR.EntropyPFR\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\n/-!\n# Polynomial Freiman-Ruzsa conjecture\n\nHere we prove the polynomial Freiman-Ruzsa conjecture.\n-/\n\nopen ProbabilityTheory MeasureTheory Real Set Fintype Function\nopen scoped BigOperators Pointwise\n\nuniverse u\n\nnamespace ProbabilityTheory\nvariable {G \u03a9 : Type*} [AddCommGroup G] [Fintype G]\n    [MeasurableSpace G] [MeasurableSingletonClass G] {A B : Finset G}\n    [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U V : \u03a9 \u2192 G}\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V` with probability `# (A \u2229 B) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub_zero (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {0})\n      = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n  have : (U - V) \u207b\u00b9' {0} = \u22c3 (g : G), (U \u207b\u00b9' {g} \u2229 V\u207b\u00b9' {g}) := by\n    ext \u03c9; simp [sub_eq_zero, eq_comm]\n  rw [this, measureReal_iUnion_fintype _\n    (fun i \u21a6 (Umeas $ measurableSet_discrete _).inter $ Vmeas $ measurableSet_discrete _)]; swap\n  \u00b7 intro g g' hgg'\n    apply Set.disjoint_iff_inter_eq_empty.2\n    ext a\n    simp (config := {contextual := True}) [hgg']\n  classical\n  let W : Finset G := A \u2229 B\n  calc\n    \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p} \u2229 V \u207b\u00b9' {p})\n      = \u2211 p, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply sum_congr _ _ (fun g \u21a6 ?_)\n        rw [hindep.measureReal_inter_preimage_eq_mul (measurableSet_discrete _) $\n          measurableSet_discrete _]\n    _ = \u2211 p in W, (\u2119 : Measure \u03a9).real (U \u207b\u00b9' {p}) * (\u2119 : Measure \u03a9).real (V \u207b\u00b9' {p}) := by\n        apply (Finset.sum_subset W.subset_univ _).symm\n        intro i _ hi\n        replace hi : i \u2209 A \u2228 i \u2209 B := by simp [W] at hi; tauto\n        rcases hi with h'i|h'i\n        \u00b7 simp [Uunif.measureReal_preimage_of_nmem h'i]\n        \u00b7 simp [Vunif.measureReal_preimage_of_nmem h'i]\n    _ = \u2211 p in W, (1 / Nat.card A : \u211d) * (1 / Nat.card B) := by\n        apply Finset.sum_congr rfl (fun i hi \u21a6 ?_)\n        replace hi : i \u2208 A \u2227 i \u2208 B := by simpa [W] using hi\n        rw [Uunif.measureReal_preimage_of_mem (by trivial) hi.1,\n            Vunif.measureReal_preimage_of_mem (by trivial) hi.2]\n    _ = (W.card : \u211d) / (Nat.card A * Nat.card B) := by simp [div_eq_inv_mul]; ring\n    _ = Nat.card (A \u2229 B : Set G) / (Nat.card A * Nat.card B) := by\n        congr\n        rw [\u2190 Finset.coe_inter, Nat.card_eq_fintype_card, Fintype.card_ofFinset]\n        simp\n\n/-- Given two independent random variables `U` and `V` uniformly distributed respectively on `A`\nand `B`, then `U = V + x` with probability `# (A \u2229 (B + x)) / #A \u2b1d #B`. -/\nlemma IsUniform.measureReal_preimage_sub (Uunif : IsUniform A U) (Umeas : Measurable U)\n    (Vunif : IsUniform B V) (Vmeas : Measurable V) (hindep : IndepFun U V) (x : G) :\n    (\u2119 : Measure \u03a9).real ((U - V) \u207b\u00b9' {x})\n      = Nat.card (A \u2229 (B + {x}) : Set G) / (Nat.card A * Nat.card B) := by\n  classical\n  let W := fun \u03c9 \u21a6 V \u03c9 + x\n  have Wunif : IsUniform (B + {x} : Set G) W := by\n    convert Vunif.comp (add_left_injective x)\n    simp\n  have Wmeas : Measurable W := Vmeas.add_const _\n  have UWindep : IndepFun U W := by\n    have : Measurable (fun g \u21a6 g + x) := measurable_add_const x\n    exact hindep.comp measurable_id this\n  have : (U - V) \u207b\u00b9' {x} = (U - W) \u207b\u00b9' {0} := by\n    ext \u03c9\n    simp only [W, mem_preimage, Pi.add_apply, mem_singleton_iff, Pi.sub_apply, \u2190 sub_eq_zero (b := x)]\n    abel_nf\n  have h : (B:Set G)+{x} = (B+{x}:Finset G) := by simp\n  rw [h] at Wunif\n  rw [this, Uunif.measureReal_preimage_sub_zero Umeas Wunif Wmeas UWindep]\n  congr 3\n  . rw [add_singleton]; simp\n  convert Finset.card_vadd_finset (AddOpposite.op x) B\n  . simp\n  simp\n\nend ProbabilityTheory\n\n\n/-- Record positivity results that are useful in the proof of PFR. -/\nlemma PFR_conjecture_pos_aux {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A - A) \u2227 0 < K := by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A - A) := by\n    have : Nonempty (A - A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.sub h\u2080A h\u2080A)\n    have : Finite (A - A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nlemma PFR_conjecture_pos_aux' {G : Type*} [AddCommGroup G] {A : Set G} [Finite A] {K : \u211d} (h\u2080A : A.Nonempty) (hA : Nat.card (A + A) \u2264 K * Nat.card A) :\n    (0 : \u211d) < Nat.card A \u2227 (0 : \u211d) < Nat.card (A + A) \u2227 0 < K := by\n  have card_AA_pos : (0 : \u211d) < Nat.card (A + A) := by\n    have : Nonempty (A + A) := Set.nonempty_coe_sort.mpr (Set.Nonempty.add h\u2080A h\u2080A)\n    have : Finite (A + A) := finite_coe_iff.mpr (Finite.image2 _ (Set.toFinite A) (Set.toFinite A))\n    simp [Nat.cast_pos, Nat.card_pos_iff]\n  have KA_pos : 0 < K \u2227 (0 : \u211d) < Nat.card A := by\n    have I : \u00ac ((Nat.card A : \u211d) < 0) := by simp\n    simpa [Nat.cast_pos, I, and_false, or_false] using mul_pos_iff.1 (card_AA_pos.trans_le hA)\n  exact \u27e8KA_pos.2, card_AA_pos, KA_pos.1\u27e9\n\nvariable {G : Type*} [AddCommGroup G] [MeasurableSpace G]\n  [MeasurableSingletonClass G] {A : Set G} [Finite A] {K : \u211d} [Countable G]\n\n/-- A uniform distribution on a set with doubling constant `K` has self Rusza distance\nat most `log K`. -/\ntheorem rdist_le_of_isUniform_of_card_add_le (h\u2080A : A.Nonempty) (hA : Nat.card (A - A) \u2264 K * Nat.card A)\n    {\u03a9 : Type*} [MeasureSpace \u03a9] [IsProbabilityMeasure (\u2119 : Measure \u03a9)] {U\u2080 : \u03a9 \u2192 G}\n    (U\u2080unif : IsUniform A U\u2080) (U\u2080meas : Measurable U\u2080) : d[U\u2080 # U\u2080] \u2264 log K := by\n  obtain \u27e8A_pos, AA_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  rcases independent_copies_two U\u2080meas U\u2080meas with \u27e8\u03a9, m\u03a9, U, U', hP, hU, hU', UU'_indep, idU, idU'\u27e9\n  have Uunif : IsUniform A U := U\u2080unif.of_identDistrib idU.symm $ measurableSet_discrete _\n  have U'unif : IsUniform A U' := U\u2080unif.of_identDistrib idU'.symm $ measurableSet_discrete _\n  have IU : d[U # U'] \u2264 log K := by\n    have I : H[U - U'] \u2264 log (Nat.card (A - A)) := by\n      convert entropy_le_log_card_of_mem (A := (A-A).toFinite.toFinset) ?_ ?_ with x\n      . simp\n        exact Iff.rfl\n      . measurability\n      filter_upwards [Uunif.ae_mem, U'unif.ae_mem] with \u03c9 h1 h2\n      simp\n      exact Set.sub_mem_sub h1 h2\n    have J : log (Nat.card (A - A)) \u2264 log K + log (Nat.card A) := by\n      apply (log_le_log AA_pos hA).trans (le_of_eq _)\n      rw [log_mul K_pos.ne' A_pos.ne']\n--    have : H[U + U'] = H[U - U'] := by congr; simp\n    rw [UU'_indep.rdist_eq hU hU', IsUniform.entropy_eq' Uunif hU, IsUniform.entropy_eq' U'unif hU']\n    linarith\n  rwa [idU.rdist_eq idU'] at IU\n\nvariable [ElementaryAddCommGroup G 2] [Fintype G]\n\nlemma sumset_eq_sub : A + A = A - A := by\n  rw [\u2190 Set.image2_add, \u2190 Set.image2_sub]\n  congr! 1 with a _ b _\n  show a + b = a - b\n  simp\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 $K^{13/2} |A|^{1/2} / |H|^{1/2}$ cosets of $H$, and $H$ has\nthe same cardinality as $A$ up to a multiplicative factor $K^11$. -/\nlemma PFR_conjecture_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 ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n      \u2227 Nat.card H \u2264 K ^ 11 * Nat.card A \u2227 Nat.card A \u2264 K ^ 11 * Nat.card H \u2227 A \u2286 c + H := by\n  classical\n  let _mG : MeasurableSpace G := \u22a4\n  rw [sumset_eq_sub] at hA\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 : d[UA # UA] \u2264 log K := rdist_le_of_isUniform_of_card_add_le h\u2080A hA UAunif UAmeas\n  rw [\u2190 sumset_eq_sub] at hA\n  let p : refPackage \u03a9\u2080 \u03a9\u2080 G := \u27e8UA, UA, UAmeas, UAmeas, 1/9, (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 p (by norm_num) 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 11/2 * 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 11 * 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 ^ 11 * Nat.card (H : Set G) := by\n    have : log (Nat.card A) \u2264 log K * 11 + 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 ^ 11 * Nat.card A := by\n    have : log (Nat.card (H : Set G)) \u2264 log K * 11 + 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 * (-11/2) + 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 ^ (-11/2) * (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 ^ (13/2) * (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 ^ (11/2) * 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    rw [add_sub_assoc] at Au\n    refine Au.trans $ add_subset_add_left $\n      (sub_subset_sub (inter_subset_right ..) (inter_subset_right ..)).trans ?_\n    rw [add_sub_add_comm, singleton_sub_singleton, sub_self]\n    simp\n  exact \u27e8H, u, Iu, IHA, IAH, A_subset_uH\u27e9\n\n\n/-- 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 $2K^{12}$ cosets of\na subgroup of cardinality at most $|A|$. -/\ntheorem PFR_conjecture (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 ^ 12 \u2227 Nat.card H \u2264 Nat.card A \u2227 A \u2286 c + H := 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 ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2)\n      \u2227 Nat.card (H : Set G) \u2264 K ^ 11 * Nat.card A \u2227 Nat.card A \u2264 K ^ 11 * Nat.card (H : Set G)\n      \u2227 A \u2286 c + H :=\n    PFR_conjecture_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 ^ (13/2) * (Nat.card A) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := hc\n    _ \u2264 K ^ (13/2) * (K ^ 11 * Nat.card (H : Set G)) ^ (1/2) * (Nat.card (H : Set G)) ^ (-1/2) := by\n      gcongr\n    _ = K ^ 12 := by rpow_ring; norm_num\n    _ < 2 * K ^ 12 := by linarith [show 0 < K ^ 12 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 ^ (13/2) * (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 ^ (13/2) * (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 ^ (13/2) * (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 ^ (13/2) * (Nat.card A) ^ (-1/2) * (K ^ 11 * Nat.card A) ^ (1/2) := by\n        gcongr\n    _ = 2 * K ^ 12 := by\n        rpow_ring\n        norm_num\n\n", "theoremStatement": "/-- Corollary of `PFR_conjecture` in which the ambient group is not required to be finite (but) then\n$H$ and $c$ are finite. -/\ntheorem PFR_conjecture' {G : Type*} [AddCommGroup G] [ElementaryAddCommGroup G 2]\n    {A : Set G} {K : \u211d} (h\u2080A : A.Nonempty) (Afin : A.Finite)\n    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"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"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  let G' := AddSubgroup.closure A\n  let G'fin : Fintype G' := by\n    exact Finite.fintype (ElementaryAddCommGroup.finite_closure Afin)\n  have G'Elem : ElementaryAddCommGroup G' 2 := ElementaryAddCommGroup.subgroup _\n  let \u03b9 : G'\u2192+ G := G'.subtype\n  have \u03b9_inj : Injective \u03b9 := AddSubgroup.subtype_injective G'\n  let A' : Set G' := \u03b9 \u207b\u00b9' A\n  have A_rg : A \u2286 range \u03b9 := by\n    simp only [AddSubgroup.coeSubtype, Subtype.range_coe_subtype, SetLike.mem_coe, G', \u03b9]\n    exact AddSubgroup.subset_closure\n  have cardA' : Nat.card A' = Nat.card A := Nat.card_preimage_of_injective \u03b9_inj A_rg\n  have hA' : Nat.card (A' + A') \u2264 K * Nat.card A' := by\n    rwa [cardA', \u2190 preimage_add _ \u03b9_inj A_rg A_rg,\n         Nat.card_preimage_of_injective \u03b9_inj (add_subset_range _ A_rg A_rg)]\n  rcases PFR_conjecture (h\u2080A.preimage' A_rg) hA' with \u27e8H', c', hc', hH', hH'\u2082\u27e9\n  refine \u27e8AddSubgroup.map \u03b9 H', \u03b9 '' c', toFinite _, toFinite (\u03b9 '' H'), ?_, ?_, fun x hx \u21a6 ?_\u27e9\n  \u00b7 rwa [Nat.card_image_of_injective \u03b9_inj]\n  \u00b7 erw [Nat.card_image_of_injective \u03b9_inj, \u2190 cardA']\n    exact hH'\n  \u00b7 erw [\u2190 image_add]\n    exact \u27e8\u27e8x, AddSubgroup.subset_closure hx\u27e9, hH'\u2082 hx, rfl\u27e9", "proofType": "tactic", "proofLengthLines": 21, "proofLengthTokens": 1144}}
{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n  [MeasurableSingletonClass G] [elem: 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/-- `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", "theoremStatement": "/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n    + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k ", "theoremName": "rdist_add_rdist_add_condMutual_eq", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "2e4cc62", "date": "2023-12-01"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": 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"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", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n  have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n  have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n  have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n  have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n    (fun i => by fin_cases i <;> assumption)\n  rw [h0, h1, h2, h3] at h\n  have heq : d[X\u2082' # X\u2081'] = k := by\n    rw [rdist_symm]\n    apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n  rw [heq] at h\n  convert h.symm using 1\n  \u00b7 congr 2 <;> abel\n  \u00b7 ring", "proofType": "tactic", "proofLengthLines": 14, "proofLengthTokens": 519}}
{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n  [MeasurableSingletonClass G] [elem: 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/-- `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/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n    + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n  have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n  have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n  have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n  have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n  have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n    (fun i => by fin_cases i <;> assumption)\n  rw [h0, h1, h2, h3] at h\n  have heq : d[X\u2082' # X\u2081'] = k := by\n    rw [rdist_symm]\n    apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n  rw [heq] at h\n  convert h.symm using 1\n  \u00b7 congr 2 <;> abel\n  \u00b7 ring\n\n", "theoremStatement": "/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n    d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) ", "theoremName": "rdist_of_sums_ge", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 72, "tokenPositionInFile": 2571, "theoremPositionInFile": 1}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 7, "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", 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"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", 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{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n  [MeasurableSingletonClass G] [elem: 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/-- `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/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n    + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n  have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n  have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n  have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n  have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n  have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n    (fun i => by fin_cases i <;> assumption)\n  rw [h0, h1, h2, h3] at h\n  have heq : d[X\u2082' # X\u2081'] = k := by\n    rw [rdist_symm]\n    apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n  rw [heq] at h\n  convert h.symm using 1\n  \u00b7 congr 2 <;> abel\n  \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n    d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n", "theoremStatement": "/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n    d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) ", "theoremName": "condRuzsaDist_of_sums_ge", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 80, "tokenPositionInFile": 2972, "theoremPositionInFile": 2}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 9, 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"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", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":=\n  condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)", "proofType": "term", "proofLengthLines": 1, "proofLengthTokens": 90}}
{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n  [MeasurableSingletonClass G] [elem: 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/-- `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/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n    + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n  have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n  have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n  have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n  have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n  have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n    (fun i => by fin_cases i <;> assumption)\n  rw [h0, h1, h2, h3] at h\n  have heq : d[X\u2082' # X\u2081'] = k := by\n    rw [rdist_symm]\n    apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n  rw [heq] at h\n  convert h.symm using 1\n  \u00b7 congr 2 <;> abel\n  \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n    d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n    d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n", "theoremStatement": "/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 ", "theoremName": "diff_rdist_le_1", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "0ec25b1", "date": 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"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", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n  convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n  \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n  \u00b7 exact h\u2082.entropy_eq", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 275}}
{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n  [MeasurableSingletonClass G] [elem: 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/-- `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/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n    + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n  have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n  have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n  have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n  have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n  have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n    (fun i => by fin_cases i <;> assumption)\n  rw [h0, h1, h2, h3] at h\n  have heq : d[X\u2082' # X\u2081'] = k := by\n    rw [rdist_symm]\n    apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n  rw [heq] at h\n  convert h.symm using 1\n  \u00b7 congr 2 <;> abel\n  \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n    d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n    d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n  have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n  convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n  \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n  \u00b7 exact h\u2082.entropy_eq\n\n", "theoremStatement": "/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_2 : d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 ", "theoremName": "diff_rdist_le_2", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "96a97b5", "date": "2023-11-27"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 96, "tokenPositionInFile": 3939, "theoremPositionInFile": 4}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 97, "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", 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"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", 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"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", 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"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", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n  convert condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 4\n  \u00b7 rw [rdist_symm]\n    exact (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081\n  \u00b7 exact h\u2081.entropy_eq", "proofType": "tactic", "proofLengthLines": 5, "proofLengthTokens": 264}}
{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n  [MeasurableSingletonClass G] [elem: 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/-- `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/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n    + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n  have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n  have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n  have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n  have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n  have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n    (fun i => by fin_cases i <;> assumption)\n  rw [h0, h1, h2, h3] at h\n  have heq : d[X\u2082' # X\u2081'] = k := by\n    rw [rdist_symm]\n    apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n  rw [heq] at h\n  convert h.symm using 1\n  \u00b7 congr 2 <;> abel\n  \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n    d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n    d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n  have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n  convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n  \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n  \u00b7 exact h\u2082.entropy_eq\n\n/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_2 : d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n  have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n  convert condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 4\n  \u00b7 rw [rdist_symm]\n    exact (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081\n  \u00b7 exact h\u2081.entropy_eq\n\n", "theoremStatement": "/-- $$ d[X_1^0;X_1|X_1+\\tilde X_2] - d[X_1^0;X_1] \\leq\n    \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_3 : d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 ", "theoremName": "diff_rdist_le_3", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "ba82447", "date": "2023-11-28"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 104, "tokenPositionInFile": 4426, "theoremPositionInFile": 5}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 11, "numPremises": 96, "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", 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"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", 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"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", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n  convert condRuzsaDist_diff_le''' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 3\n  \u00b7 rw [(IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082]\n  \u00b7 apply h\u2082.entropy_eq", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 245}}
{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n  [MeasurableSingletonClass G] [elem: 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/-- `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/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n    + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n  have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n  have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n  have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n  have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n  have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n    (fun i => by fin_cases i <;> assumption)\n  rw [h0, h1, h2, h3] at h\n  have heq : d[X\u2082' # X\u2081'] = k := by\n    rw [rdist_symm]\n    apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n  rw [heq] at h\n  convert h.symm using 1\n  \u00b7 congr 2 <;> abel\n  \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n    d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n    d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n  have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n  convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n  \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n  \u00b7 exact h\u2082.entropy_eq\n\n/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_2 : d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n  have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n  convert condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 4\n  \u00b7 rw [rdist_symm]\n    exact (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081\n  \u00b7 exact h\u2081.entropy_eq\n\n/-- $$ d[X_1^0;X_1|X_1+\\tilde X_2] - d[X_1^0;X_1] \\leq\n    \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_3 : d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n  have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n  convert condRuzsaDist_diff_le''' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 3\n  \u00b7 rw [(IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082]\n  \u00b7 apply h\u2082.entropy_eq\n\n", "theoremStatement": "/-- $$ d[X_2^0; X_2|X_2+\\tilde X_1] - d[X_2^0; X_2] \\leq\n    \\tfrac{1}{2}k + \\tfrac{1}{4} \\mathbb{H}[X_2] - \\tfrac{1}{4} \\mathbb{H}[X_1].$$ -/\nlemma diff_rdist_le_4 : d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 ", "theoremName": "diff_rdist_le_4", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "ba82447", "date": "2023-11-28"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 112, "tokenPositionInFile": 4906, "theoremPositionInFile": 6}, "dependencyMetadata": {"inFilePremises": false, "numInFilePremises": 0, "repositoryPremises": true, "numRepositoryPremises": 12, "numPremises": 98, "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", 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"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", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n  convert condRuzsaDist_diff_le''' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 3\n  \u00b7 rw [rdist_symm, (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081]\n  \u00b7 apply h\u2081.entropy_eq", "proofType": "tactic", "proofLengthLines": 4, "proofLengthTokens": 257}}
{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n  [MeasurableSingletonClass G] [elem: 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/-- `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/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n    + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n  have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n  have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n  have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n  have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n  have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n    (fun i => by fin_cases i <;> assumption)\n  rw [h0, h1, h2, h3] at h\n  have heq : d[X\u2082' # X\u2081'] = k := by\n    rw [rdist_symm]\n    apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n  rw [heq] at h\n  convert h.symm using 1\n  \u00b7 congr 2 <;> abel\n  \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n    d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n    d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n  have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n  convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n  \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n  \u00b7 exact h\u2082.entropy_eq\n\n/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_2 : d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n  have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n  convert condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 4\n  \u00b7 rw [rdist_symm]\n    exact (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081\n  \u00b7 exact h\u2081.entropy_eq\n\n/-- $$ d[X_1^0;X_1|X_1+\\tilde X_2] - d[X_1^0;X_1] \\leq\n    \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_3 : d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n  have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n  convert condRuzsaDist_diff_le''' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 3\n  \u00b7 rw [(IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082]\n  \u00b7 apply h\u2082.entropy_eq\n\n/-- $$ d[X_2^0; X_2|X_2+\\tilde X_1] - d[X_2^0; X_2] \\leq\n    \\tfrac{1}{2}k + \\tfrac{1}{4} \\mathbb{H}[X_2] - \\tfrac{1}{4} \\mathbb{H}[X_1].$$ -/\nlemma diff_rdist_le_4 : d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n  have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n  convert condRuzsaDist_diff_le''' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 3\n  \u00b7 rw [rdist_symm, (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081]\n  \u00b7 apply h\u2081.entropy_eq\n\n", "theoremStatement": "/-- We have $I_1 \\leq 2 \\eta k$ -/\nlemma first_estimate : I\u2081 \u2264 2 * p.\u03b7 * k ", "theoremName": "first_estimate", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 120, "tokenPositionInFile": 5399, "theoremPositionInFile": 7}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 7, "repositoryPremises": true, "numRepositoryPremises": 19, "numPremises": 239, "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", 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"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", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  have v1 := rdist_add_rdist_add_condMutual_eq X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a\n  have v2 := rdist_of_sums_ge p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a\n  have v3 := condRuzsaDist_of_sums_ge p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a (by measurability) (by measurability)\n  have v4 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_1 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n  have v5 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_2 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n  have v6 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_3 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n  have v7 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_4 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n  simp only [inv_eq_one_div] at *\n  linarith [v1, v2, v3, v4, v5, v6, v7]", "proofType": "tactic", "proofLengthLines": 9, "proofLengthTokens": 690}}
{"srcContext": "import PFR.Fibring\nimport PFR.TauFunctional\n\n/-!\n# First estimate\n\nThe first estimate on tau-minimizers.\n\nAssumptions:\n\n* $X_1, X_2$ are tau-minimizers\n* $X_1, X_2, \\tilde X_1, \\tilde X_2$ are independent random variables, with $X_1,\\tilde X_1$ copies of $X_1$ and $X_2,\\tilde X_2$ copies of $X_2$.\n* $k := d[X_1;X_2]$\n* $I_1 := I [X_1+X_2 : \\tilde X_1 + X_2 | X_1+X_2+\\tilde X_1+\\tilde X_2]$\n\n## Main results\n\n* `first_estimate` : $I_1 \u2264 2 \u03b7 k$\n* `ent_ofsum_le` : $H[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} H[X_1]+\\tfrac{1}{2} H[X_2] + (2 + \\eta) k - I_1.$\n-/\n\nopen MeasureTheory ProbabilityTheory\n\nvariable {G : Type*} [addgroup: AddCommGroup G] [Fintype G] [hG : MeasurableSpace G]\n  [MeasurableSingletonClass G] [elem: 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/-- `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/-- The sum of\n$$ d[X_1+\\tilde X_2;X_2+\\tilde X_1] + d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1] $$\nand\n$$ I[X_1+ X_2 : \\tilde X_1 + X_2 \\,|\\, X_1 + X_2 + \\tilde X_1 + \\tilde X_2] $$\nis equal to $2k$. -/\nlemma rdist_add_rdist_add_condMutual_eq : d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] + d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n    + I[X\u2081 + X\u2082 : X\u2081' + X\u2082 | X\u2081 + X\u2082 + X\u2081' + X\u2082'] = 2 * k := by\n  have h0 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 0 = X\u2081 := rfl\n  have h1 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 1 = X\u2082 := rfl\n  have h2 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 2 = X\u2082' := rfl\n  have h3 : ![X\u2081, X\u2082, X\u2082', X\u2081'] 3 = X\u2081' := rfl\n  have h := sum_of_rdist_eq_char_2 ![X\u2081, X\u2082, X\u2082', X\u2081'] h_indep\n    (fun i => by fin_cases i <;> assumption)\n  rw [h0, h1, h2, h3] at h\n  have heq : d[X\u2082' # X\u2081'] = k := by\n    rw [rdist_symm]\n    apply ProbabilityTheory.IdentDistrib.rdist_eq h\u2081.symm h\u2082.symm\n  rw [heq] at h\n  convert h.symm using 1\n  \u00b7 congr 2 <;> abel\n  \u00b7 ring\n\n/-- The distance $d[X_1+\\tilde X_2; X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1]) - \\eta (d[X^0_2; X_2+\\tilde X_1] - d[X^0_2; X_2]).$$ -/\nlemma rdist_of_sums_ge :\n    d[X\u2081 + X\u2082' # X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  distance_ge_of_min _ h_min (hX\u2081.add hX\u2082') (hX\u2082.add hX\u2081')\n\n/-- The distance $d[X_1|X_1+\\tilde X_2; X_2|X_2+\\tilde X_1]$ is at least\n$$ k - \\eta (d[X^0_1; X_1 | X_1 + \\tilde X_2] - d[X^0_1; X_1]) - \\eta(d[X^0_2; X_2 | X_2 + \\tilde X_1] - d[X^0_2; X_2]).$$\n-/\nlemma condRuzsaDist_of_sums_ge :\n    d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081'] \u2265\n      k - p.\u03b7 * (d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081])\n        - p.\u03b7 * (d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082]) :=\n  condRuzsaDistance_ge_of_min _ h_min hX\u2081 hX\u2082 _ _ (by measurability) (by measurability)\n\n/-- $$d[X^0_1; X_1+\\tilde X_2] - d[X^0_1; X_1] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\bbH[X_2] - \\tfrac{1}{4} \\bbH[X_1].$$ -/\nlemma diff_rdist_le_1 : d[p.X\u2080\u2081 # X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n  have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n  convert condRuzsaDist_diff_le' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 4\n  \u00b7 exact ProbabilityTheory.IdentDistrib.rdist_eq (IdentDistrib.refl hX\u2081.aemeasurable) h\u2082\n  \u00b7 exact h\u2082.entropy_eq\n\n/-- $$ d[X^0_2;X_2+\\tilde X_1] - d[X^0_2; X_2] \\leq \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_2 : d[p.X\u2080\u2082 # X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n  have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n  convert condRuzsaDist_diff_le' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 4\n  \u00b7 rw [rdist_symm]\n    exact (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081\n  \u00b7 exact h\u2081.entropy_eq\n\n/-- $$ d[X_1^0;X_1|X_1+\\tilde X_2] - d[X_1^0;X_1] \\leq\n    \\tfrac{1}{2} k + \\tfrac{1}{4} \\mathbb{H}[X_1] - \\tfrac{1}{4} \\mathbb{H}[X_2].$$ -/\nlemma diff_rdist_le_3 : d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082'] - d[p.X\u2080\u2081 # X\u2081] \u2264 k/2 + H[X\u2081]/4 - H[X\u2082]/4 := by\n  have h : IndepFun X\u2081 X\u2082' := by simpa using h_indep.indepFun (show (0:Fin 4) \u2260 2 by decide)\n  convert condRuzsaDist_diff_le''' \u2119 p.hmeas1 hX\u2081 hX\u2082' h using 3\n  \u00b7 rw [(IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082]\n  \u00b7 apply h\u2082.entropy_eq\n\n/-- $$ d[X_2^0; X_2|X_2+\\tilde X_1] - d[X_2^0; X_2] \\leq\n    \\tfrac{1}{2}k + \\tfrac{1}{4} \\mathbb{H}[X_2] - \\tfrac{1}{4} \\mathbb{H}[X_1].$$ -/\nlemma diff_rdist_le_4 : d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081'] - d[p.X\u2080\u2082 # X\u2082] \u2264 k/2 + H[X\u2082]/4 - H[X\u2081]/4 := by\n  have h : IndepFun X\u2082 X\u2081' := by simpa using h_indep.indepFun (show (1:Fin 4) \u2260 3 by decide)\n  convert condRuzsaDist_diff_le''' \u2119 p.hmeas2 hX\u2082 hX\u2081' h using 3\n  \u00b7 rw [rdist_symm, (IdentDistrib.refl hX\u2082.aemeasurable).rdist_eq h\u2081]\n  \u00b7 apply h\u2081.entropy_eq\n\n/-- We have $I_1 \\leq 2 \\eta k$ -/\nlemma first_estimate : I\u2081 \u2264 2 * p.\u03b7 * k := by\n  have v1 := rdist_add_rdist_add_condMutual_eq X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a\n  have v2 := rdist_of_sums_ge p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a\n  have v3 := condRuzsaDist_of_sums_ge p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a (by measurability) (by measurability)\n  have v4 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_1 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n  have v5 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_2 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n  have v6 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_3 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n  have v7 := (mul_le_mul_left p.h\u03b7).2 (diff_rdist_le_4 p X\u2081 X\u2082 X\u2081' X\u2082' \u2039_\u203a \u2039_\u203a \u2039_\u203a \u2039_\u203a)\n  simp only [inv_eq_one_div] at *\n  linarith [v1, v2, v3, v4, v5, v6, v7]\n\n", "theoremStatement": "/--\n$$\\mathbb{H}[X_1+X_2+\\tilde X_1+\\tilde X_2] \\le \\tfrac{1}{2} \\mathbb{H}[X_1]+\\tfrac{1}{2} \\mathbb{H}[X_2] + (2 + \\eta) k - I_1.$$\n-/\nlemma ent_ofsum_le : H[X\u2081 + X\u2082 + X\u2081' + X\u2082'] \u2264 H[X\u2081]/2 + H[X\u2082]/2 + (2+p.\u03b7)*k - I\u2081 ", "theoremName": "ent_ofsum_le", "fileCreated": {"commit": "f3d2e40", "date": "2023-12-02"}, "theoremCreated": {"commit": "29a4d44", "date": "2023-12-09"}, "file": "PFR/PFR/FirstEstimate.lean", "module": "PFR.FirstEstimate", "jsonFile": "PFR.FirstEstimate.jsonl", "positionMetadata": {"lineInFile": 132, "tokenPositionInFile": 6166, "theoremPositionInFile": 8}, "dependencyMetadata": {"inFilePremises": true, "numInFilePremises": 4, "repositoryPremises": true, "numRepositoryPremises": 21, "numPremises": 340, "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", 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"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", "PFR.ForMathlib.Entropy.RuzsaDist", "PFR.Fibring", "PFR.TauFunctional"]}, "proofMetadata": {"hasProof": true, "proof": ":= by\n  let D := d[X\u2081 + X\u2082' # X\u2082 + X\u2081']\n  let Dcc := d[X\u2081 | X\u2081 + X\u2082' # X\u2082 | X\u2082 + X\u2081']\n  let D1 := d[p.X\u2080\u2081 # X\u2081]\n  let Dc1 := d[p.X\u2080\u2081 # X\u2081 | X\u2081 + X\u2082']\n  let D2 := d[p.X\u2080\u2082 # X\u2082]\n  let Dc2 := d[p.X\u2080\u2082 # X\u2082 | X\u2082 + X\u2081']\n  have lem68 : D + Dcc + I\u2081 = 2 * k :=\n    rdist_add_rdist_add_condMutual_eq _ _ _ _ hX\u2081 hX\u2082 hX\u2081' hX\u2082' h\u2081 h\u2082 h_indep\n  have lem610 : Dcc \u2265 k - p.\u03b7 * (Dc1 - D1) - p.\u03b7 * (Dc2 - D2) :=\n    condRuzsaDist_of_sums_ge p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082 (by measurability) (by measurability) h_min\n  have lem611c : Dc1 - D1 \u2264 k / 2 + H[X\u2081] / 4 - H[X\u2082] / 4 :=\n    diff_rdist_le_3 p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2081 hX\u2082' h\u2082 h_indep\n  have lem611d : Dc2 - D2 \u2264 k / 2 + H[X\u2082] / 4 - H[X\u2081] / 4 :=\n    diff_rdist_le_4 p X\u2081 X\u2082 X\u2081' X\u2082' hX\u2082 hX\u2081' h\u2081 h_indep\n  have aux : D + I\u2081 \u2264 (1 + p.\u03b7) * k := by\n    calc D + I\u2081\n       \u2264 k + p.\u03b7 * (Dc1 - D1) + p.\u03b7 * (Dc2 - D2)                                         := ?_\n     _ \u2264 k + p.\u03b7 * (k / 2 + H[X\u2081] / 4 - H[X\u2082] / 4) + p.\u03b7 * (k / 2 + H[X\u2082] / 4 - H[X\u2081] / 4)  := ?_\n     _ = (1 + p.\u03b7) * k                                                                    := by ring\n    \u00b7 convert add_le_add lem68.le (neg_le_neg lem610) using 1 <;> ring\n    \u00b7 refine add_le_add (add_le_add (le_refl _) ?_) ?_\n      \u00b7 apply (mul_le_mul_left p.h\u03b7).mpr lem611c\n      \u00b7 apply (mul_le_mul_left p.h\u03b7).mpr lem611d\n  have ent_sub_eq_ent_add : H[X\u2081 + X\u2082' - (X\u2082 + X\u2081')] = H[X\u2081 + X\u2082' + (X\u2082 + X\u2081')] := by simp\n  have rw\u2081 : X\u2081 + X\u2082' + (X\u2082 + X\u2081') = X\u2081 + X\u2082 + X\u2081' + X\u2082' := by abel\n  have ind_aux : IndepFun (X\u2081 + X\u2082') (X\u2082 + X\u2081') := by\n    exact iIndepFun.indepFun_add_add h_indep (fun i \u21a6 by fin_cases i <;> assumption) 0 2 1 3\n      (by decide) (by decide) (by decide) (by decide)\n  have ind : D = H[X\u2081 + X\u2082' - (X\u2082 + X\u2081')] - H[X\u2081 + X\u2082'] / 2 - H[X\u2082 + X\u2081'] / 2 :=\n    @IndepFun.rdist_eq \u03a9 G _ \u2119 _ _ _ (X\u2081 + X\u2082') _ (X\u2082 + X\u2081') ind_aux (by measurability) (by measurability)\n  rw [ind, ent_sub_eq_ent_add, rw\u2081] at aux\n  have obs : H[X\u2081 + X\u2082 + X\u2081' + X\u2082'] \u2264 H[X\u2081 + X\u2082'] / 2 + H[X\u2082 + X\u2081'] / 2 + (1 + p.\u03b7) * k - I\u2081 := by\n    linarith\n  have rw\u2082 : H[X\u2081 + X\u2082'] = k + H[X\u2081]/2 + H[X\u2082]/2 := by\n    have HX\u2082_eq : H[X\u2082] = H[X\u2082'] :=\n      congr_arg (fun (\u03bc : Measure G) \u21a6 measureEntropy (\u03bc := \u03bc)) h\u2082.map_eq\n    have k_eq : k = H[X\u2081 - X\u2082'] - H[X\u2081] / 2 - H[X\u2082'] / 2 := by\n      have k_eq_aux : k = d[X\u2081 # X\u2082'] := (IdentDistrib.refl hX\u2081.aemeasurable).rdist_eq h\u2082\n      rw [k_eq_aux]\n      exact (h_indep.indepFun (show (0 : Fin 4) \u2260 2 by decide)).rdist_eq hX\u2081 hX\u2082'\n    rw [k_eq, \u2190 ElementaryAddCommGroup.sub_eq_add, \u2190 HX\u2082_eq]\n    ring\n  have rw\u2083 : H[X\u2082 + X\u2081'] = k + H[X\u2081]/2 + H[X\u2082]/2 := by\n    have HX\u2081_eq : H[X\u2081] = H[X\u2081'] :=\n      congr_arg (fun (\u03bc : Measure G) \u21a6 measureEntropy (\u03bc := \u03bc)) h\u2081.map_eq\n    have k_eq' : k = H[X\u2081' - X\u2082] - H[X\u2081'] / 2 - H[X\u2082] / 2 := by\n      have k_eq_aux : k = d[X\u2081' # X\u2082] :=\n        IdentDistrib.rdist_eq h\u2081 (IdentDistrib.refl hX\u2082.aemeasurable)\n      rw [k_eq_aux]\n      exact IndepFun.rdist_eq (h_indep.indepFun (show (3 : Fin 4) \u2260 1 by decide)) hX\u2081' hX\u2082\n    rw [add_comm X\u2082 X\u2081', k_eq', \u2190 ElementaryAddCommGroup.sub_eq_add, \u2190 HX\u2081_eq]\n    ring\n  calc H[X\u2081 + X\u2082 + X\u2081' + X\u2082']\n      \u2264 H[X\u2081 + X\u2082'] / 2 + H[X\u2082 + X\u2081'] / 2 + (1 + p.\u03b7) * k - I\u2081    := obs\n    _ = (k + H[X\u2081] / 2 + H[X\u2082] / 2) / 2\n        + (k + H[X\u2081] / 2 + H[X\u2082] / 2) / 2 + (1 + p.\u03b7) * k - I\u2081    := by rw [rw\u2082, rw\u2083]\n    _ = H[X\u2081] / 2 + H[X\u2082] / 2 + (2 + p.\u03b7) * k - I\u2081                := by ring", "proofType": "tactic", "proofLengthLines": 57, "proofLengthTokens": 3340}}