Sync from openai/math@faee2cf485a8 (part 27)

lean/OAI.lean

@@ -1,914 +1,914 @@
-import OAI.Algebra.AffineCancellation.Main
-import OAI.Algebra.AlgebraicObstruction.AlgebraicInput
-import OAI.Algebra.AlgebraicObstruction.Holonomy
-import OAI.Algebra.AuslanderReiten.All
-import OAI.Algebra.BogomolovPop.Injectivity
-import OAI.Algebra.Cohomology.SemilinearCochains
-import OAI.Algebra.DepthFive.Main
-import OAI.Algebra.Determinantal.Main
-import OAI.Algebra.Divisor.Normalization
-import OAI.Algebra.Divisor.RationalRecovery
-import OAI.Algebra.Drinfeld.Completion
-import OAI.Algebra.FiniteTensor.Main
-import OAI.Algebra.Finitistic.Main
-import OAI.Algebra.FinitisticAsymmetry.Main
-import OAI.Algebra.FormalGroup.Honda.Main
-import OAI.Algebra.FormulaHitting.Main
-import OAI.Algebra.FrobeniusObstruction.Obstruction
-import OAI.Algebra.GroupRing.CylinderRetention
-import OAI.Algebra.GroupRing.Main
-import OAI.Algebra.MarkedTensor.Main
-import OAI.Algebra.Module.CompactPadic
-import OAI.Algebra.OddKaplansky.GeneralTransfer
-import OAI.Algebra.PolynomialChains.Real
-import OAI.Algebra.Universal.Coset
-import OAI.Algebra.Universal.GraphDecidability
-import OAI.AlgebraicGeometry.AbhyankarSathaye.Counterexample
-import OAI.AlgebraicGeometry.CartierSections
-import OAI.AlgebraicGeometry.CharacterVarieties
-import OAI.AlgebraicGeometry.CommutingDerivations.OrdinaryExtraMain
-import OAI.AlgebraicGeometry.LogKodaira.FiberNegative
-import OAI.AlgebraicGeometry.LogKodaira.RootFactor
-import OAI.AlgebraicGeometry.NumericalDimension.IndexedTransfer
-import OAI.AlgebraicGeometry.NumericalDimension.RelativeNegativity
-import OAI.AlgebraicGeometry.NumericalDimension.TwistedPushforward
-import OAI.AlgebraicGeometry.PlaneCurves.Nagata
-import OAI.AlgebraicGeometry.RelativeTriviality.RelativeTriviality
-import OAI.AlgebraicGeometry.SectionFields.Main
-import OAI.AlgebraicGeometry.Seshadri.Main
-import OAI.AlgebraicGeometry.Stability.NumericalWalls
-import OAI.AlgebraicGeometry.StableCoordinate.All
-import OAI.AlgebraicGeometry.SurfaceCones
-import OAI.Analysis.AffineBernstein.Main
-import OAI.Analysis.AllenCahn.Main
-import OAI.Analysis.AncientFlat.Main
-import OAI.Analysis.BackwardIntertwiners.AlgebraCorollary
-import OAI.Analysis.BernoulliCone.Main
-import OAI.Analysis.BiholderTransport.Main
-import OAI.Analysis.BoundedHochschild.AtomicPrimitives
-import OAI.Analysis.BoundedHochschild.CenterPowers
-import OAI.Analysis.BoundedHochschild.CentralLinearPatch
-import OAI.Analysis.BoundedHochschild.CocycleNormalParts
-import OAI.Analysis.BoundedHochschild.CommutativePrimitives
-import OAI.Analysis.BoundedHochschild.FactorFinitePaths
-import OAI.Analysis.BoundedHochschild.FactorPrimitive
-import OAI.Analysis.BoundedHochschild.FreeWords
-import OAI.Analysis.BoundedHochschild.MainResult
-import OAI.Analysis.BoundedHochschild.MatrixDisplacement
-import OAI.Analysis.BoundedHochschild.TypeIICorners
-import OAI.Analysis.BoundedHochschild.TypeIPrimitives
-import OAI.Analysis.BoundedHochschild.UnitaryLifts
-import OAI.Analysis.BoxTransport.Routing
-import OAI.Analysis.Brenier.Stability
-import OAI.Analysis.BrownMeasure.Main
-import OAI.Analysis.C0Absorption.Main
-import OAI.Analysis.CStarAlgebra.PureInfiniteness.Main
-import OAI.Analysis.CardinalInterpolation.ModTwelve
-import OAI.Analysis.CharacterCriterion.Main
-import OAI.Analysis.CircleDomains.LengthArea
-import OAI.Analysis.CircleDomains.Development
-import OAI.Analysis.Commutator.Main
-import OAI.Analysis.Conductivity.Main
-import OAI.Analysis.Cotype.Main
-import OAI.Analysis.CoulombIonization.Main
-import OAI.Analysis.CoulombRadii.Main
-import OAI.Analysis.CoulombTransport.Main
-import OAI.Analysis.Crouzeix.Main
-import OAI.Analysis.Daugavet.Main
-import OAI.Analysis.DiamondDistortion.Main
-import OAI.Analysis.DirectCrouzeix.Main
-import OAI.Analysis.DiskMaximal.Main
-import OAI.Analysis.FactorGeneration.Main
-import OAI.Analysis.FiniteFactor.Main
-import OAI.Analysis.ForestSpace.Main
-import OAI.Analysis.Fourier.HeightLayers
-import OAI.Analysis.Fourier.SphereExtension
-import OAI.Analysis.FourierExtension.LpEstimate
-import OAI.Analysis.FourierLLogL.PaperMain
-import OAI.Analysis.FreeEntropy.Main
-import OAI.Analysis.FreeGroupFactor.Main
-import OAI.Analysis.FreeWords.Main
-import OAI.Analysis.Graph.Gradients
-import OAI.Analysis.GroupDeterminants.Main
-import OAI.Analysis.HardSphere.Main
-import OAI.Analysis.HenonEmden.Main
-import OAI.Analysis.HilbertCrouzeix.Main
-import OAI.Analysis.HotSpots.Main
-import OAI.Analysis.HyperbolicCones
-import OAI.Analysis.Hyperinvariant
-import OAI.Analysis.IncompressibleFlow.AffineExtension
-import OAI.Analysis.InnerProductSpace.ComplexSolvability
-import OAI.Analysis.IntegralMeans.Main
-import OAI.Analysis.IntegralMeans.SharpEndpoints
-import OAI.Analysis.InterpolatedFactors.Interpolation
-import OAI.Analysis.IrrationalRotation.Main
-import OAI.Analysis.JiangSu.Main
-import OAI.Analysis.Kakeya.FixedScale
-import OAI.Analysis.Kaplansky.Main
-import OAI.Analysis.Kaplansky.StableFiniteness
-import OAI.Analysis.Laughlin.Main
-import OAI.Analysis.Laughlin.UniformGap
-import OAI.Analysis.LaughlinFock.Main
-import OAI.Analysis.LaughlinFock.TensorIntegrals
-import OAI.Analysis.LaughlinGap.Main
-import OAI.Analysis.LaughlinGap.ThreeAveraging
-import OAI.Analysis.LiebThirring.CoerciveInverse
-import OAI.Analysis.LiebThirring.CompactAlternative
-import OAI.Analysis.LiebThirring.ContinuousRepresentative
-import OAI.Analysis.LiebThirring.DiscreteSpectrum
-import OAI.Analysis.LiebThirring.Eigenspace
-import OAI.Analysis.LiebThirring.EigenvalueSum
-import OAI.Analysis.LiebThirring.FormCongruence
-import OAI.Analysis.LiebThirring.FreeForm
-import OAI.Analysis.LiebThirring.LocalCompactness
-import OAI.Analysis.LiebThirring.LocalWeight
-import OAI.Analysis.LiebThirring.LowestEigenvalue
-import OAI.Analysis.LiebThirring.Main
-import OAI.Analysis.LiebThirring.Multiplication
-import OAI.Analysis.LiebThirring.NegativeResolvent
-import OAI.Analysis.LiebThirring.OneState
-import OAI.Analysis.LiebThirring.OperatorForm
-import OAI.Analysis.LiebThirring.OperatorGraph
-import OAI.Analysis.LiebThirring.PotentialCompactness
-import OAI.Analysis.LiebThirring.PotentialSplit
-import OAI.Analysis.LiebThirring.Resolvent
-import OAI.Analysis.LiebThirring.SchrodingerOperator
-import OAI.Analysis.LiebThirring.SobolevBound
-import OAI.Analysis.LiebThirring.SpectralMain
-import OAI.Analysis.LiebThirring.SpectralMoment
-import OAI.Analysis.LiebThirring.WeightedCutoff
-import OAI.Analysis.LienardCycles.Main
-import OAI.Analysis.Lipschitz.TriangularLattice
-import OAI.Analysis.LipschitzEquivalence.Main
-import OAI.Analysis.LipschitzFree.Main
-import OAI.Analysis.LipschitzFree.Renorming
-import OAI.Analysis.LipschitzHilbert.Main
-import OAI.Analysis.Littlewood.Main
-import OAI.Analysis.Littlewood.FiniteFlatness
-import OAI.Analysis.LpDimension.Main
-import OAI.Analysis.Mahler
-import OAI.Analysis.Mahler.Classification
-import OAI.Analysis.MarkovType.All
-import OAI.Analysis.MassAction.Main
-import OAI.Analysis.MatrixSigning.Main
-import OAI.Analysis.MetricEntropy.Main
-import OAI.Analysis.ModularRecovery.Main
-import OAI.Analysis.MumfordShah.CompactExclusion
-import OAI.Analysis.MutuallyUnbiased.Bases.ThreeMain
-import OAI.Analysis.MutuallyUnbiased.Main
-import OAI.Analysis.Naimark.Tracial
-import OAI.Analysis.NavierStokes.VelocitySpace
-import OAI.Analysis.NefAdjoint.FractionalDensity
-import OAI.Analysis.NefAdjoint.SectionGauge
-import OAI.Analysis.NodalLength.Main
-import OAI.Analysis.Nonexpansive
-import OAI.Analysis.NuclearUltrapower.Main
-import OAI.Analysis.NumericalRange.Main
-import OAI.Analysis.PathSelection.Main
-import OAI.Analysis.PeriodicLattice.Main
-import OAI.Analysis.PlanarEndpoint.Main
-import OAI.Analysis.Polynomial.Exhaustion
-import OAI.Analysis.ProductSpaces.Main
-import OAI.Analysis.Pureness.Main
-import OAI.Analysis.Quantum.DimensionTen.Main
-import OAI.Analysis.RCDHeat.Main
-import OAI.Analysis.RecursivePotentials.EquivalentNorms
-import OAI.Analysis.RecursivePotentials.JoinedGeometry
-import OAI.Analysis.RecursivePotentials.Main
-import OAI.Analysis.RecursivePotentials.Model
-import OAI.Analysis.RecursivePotentials.TreeGeometry
-import OAI.Analysis.RecursivePotentials.VariableCompletion
-import OAI.Analysis.RecursivePotentials.VariableEstimates
-import OAI.Analysis.RecursivePotentials.VariableLp
-import OAI.Analysis.RecursivePotentials.VariableStability
-import OAI.Analysis.RecursivePotentials.ZeroRootGeometry
-import OAI.Analysis.RelativeGeneration.Main
-import OAI.Analysis.RieszRectifiability.Main
-import OAI.Analysis.RotationWeights.Main
-import OAI.Analysis.SegmentLenses.HeadRanges
-import OAI.Analysis.SegmentLenses.Main
-import OAI.Analysis.SeparableQuotients.Main
-import OAI.Analysis.SeparableQuotients.Positive.Main
-import OAI.Analysis.SignedDisk.Main
-import OAI.Analysis.SphereIsometry.Main
-import OAI.Analysis.Steinitz.Main
-import OAI.Analysis.StrictMeans.Main
-import OAI.Analysis.StructuralCrouzeix.Endpoint
-import OAI.Analysis.SymmetricDomains.Main
-import OAI.Analysis.TemperedDistribution.ConformalIdentities
-import OAI.Analysis.TraceCone.Main
-import OAI.Analysis.TracialCohomology.Vanishing
-import OAI.Analysis.TracialSplitting.Main
-import OAI.Analysis.TreePotential.Main
-import OAI.Analysis.Triangular.Main
-import OAI.Analysis.TriangularHilbert.Main
-import OAI.Analysis.Unitarizability.AllDiscrete
-import OAI.Analysis.Unitarizability.UniverseTransport
-import OAI.Analysis.VlasovMaxwell.Main
-import OAI.Analysis.WeakInfiniteness.Main
-import OAI.CategoryTheory.Globular.Main
-import OAI.CategoryTheory.ThickClosure.Main
-import OAI.CategoryTheory.Thomason.RankedFaithfulness
-import OAI.Combinatorics.Automata.Main
-import OAI.Combinatorics.BalancedRyser.Main
-import OAI.Combinatorics.BooleanFunctions.Main
-import OAI.Combinatorics.Chromatic.Main
-import OAI.Combinatorics.CliqueFree.Main
-import OAI.Combinatorics.Coboundary.Main
-import OAI.Combinatorics.CompleteCrossing.Main
-import OAI.Combinatorics.ContingencyTables.FactorialLift
-import OAI.Combinatorics.Crossing.Main
-import OAI.Combinatorics.CycleDecomposition.Main
-import OAI.Combinatorics.DiscreteConvexity.Main
-import OAI.Combinatorics.Distances.PolynomialIsolation
-import OAI.Combinatorics.EditDistance.Main
-import OAI.Combinatorics.EuclideanRamsey.Circle
-import OAI.Combinatorics.EuclideanRamsey.Main
-import OAI.Combinatorics.EuclideanRamsey.NineStatement
-import OAI.Combinatorics.EuclideanRamsey.Quadratic
-import OAI.Combinatorics.EuclideanRamsey.Spherical
-import OAI.Combinatorics.EuclideanRamsey.Transitive
-import OAI.Combinatorics.ExpectationThreshold.Main
-import OAI.Combinatorics.FiniteCircle.Main
-import OAI.Combinatorics.GotsmanLinial.Main
-import OAI.Combinatorics.GraphSpectrum.SampledChromatic
-import OAI.Combinatorics.GraphThreshold.Main
-import OAI.Combinatorics.HadwigerCounterexample.Main
-import OAI.Combinatorics.HalvingLines.Main
-import OAI.Combinatorics.Hamiltonian.Main
-import OAI.Combinatorics.Honeycomb.FiniteCylinder
-import OAI.Combinatorics.Honeycomb.PolygonMoment
-import OAI.Combinatorics.HoneycombBridges.ExactLength
-import OAI.Combinatorics.HoneycombBridges.Geometry
-import OAI.Combinatorics.HoneycombChords
-import OAI.Combinatorics.HoneycombCorridor.CorridorTransfer
-import OAI.Combinatorics.IndependentSets.Main
-import OAI.Combinatorics.InfiniteMatroid.Corollaries
-import OAI.Combinatorics.InfiniteMatroid.Main
-import OAI.Combinatorics.KMedian.Main
-import OAI.Combinatorics.KMedian.Threshold
-import OAI.Combinatorics.KMedianRecovery.Main
-import OAI.Combinatorics.KMedianRecovery.Refined.Main
-import OAI.Combinatorics.KMedianRecovery.Refined.Recovery
-import OAI.Combinatorics.KServer.Main
-import OAI.Combinatorics.ListHadwiger.Main
-import OAI.Combinatorics.MatchingCount.Main
-import OAI.Combinatorics.MatchingEntropy.Main
-import OAI.Combinatorics.Permanent.Main
-import OAI.Combinatorics.PolynomialVacuum.Main
-import OAI.Combinatorics.PortMatching.Reduction
-import OAI.Combinatorics.Progressions.Main
-import OAI.Combinatorics.Ramsey.CycleClique.Main
-import OAI.Combinatorics.Ramsey.Hypercube
-import OAI.Combinatorics.RamseyFive.Main
-import OAI.Combinatorics.StableGraphs.ExteriorBounds
-import OAI.Combinatorics.StripMass.Main
-import OAI.Combinatorics.SumProduct.Alignment.Main
-import OAI.Combinatorics.ThinTrees.AlgorithmicMain
-import OAI.Combinatorics.TriangleFace.Main
-import OAI.Combinatorics.MatchingPSD.AffineLift
-import OAI.Combinatorics.MatchingPSD.Main
-import OAI.Combinatorics.MatrixRemoval.Main
-import OAI.Combinatorics.MatroidCounting.CommonBases
-import OAI.Combinatorics.ParityLifts.AsymptoticBounds
-import OAI.Combinatorics.PerfectMatching.AlgorithmPartition
-import OAI.Combinatorics.PerfectMatching.Ascent
-import OAI.Combinatorics.PerfectMatching.CompletePairs
-import OAI.Combinatorics.PerfectMatching.Completion
-import OAI.Combinatorics.PerfectMatching.CountCorrect
-import OAI.Combinatorics.PerfectMatching.DyadicEntropy
-import OAI.Combinatorics.PerfectMatching.DyadicEnvelope
-import OAI.Combinatorics.PerfectMatching.ElementaryCount
-import OAI.Combinatorics.PerfectMatching.EntropyUpper
-import OAI.Combinatorics.PerfectMatching.FeasibilityRepair
-import OAI.Combinatorics.PerfectMatching.ForcedReduction
-import OAI.Combinatorics.PerfectMatching.Main
-import OAI.Combinatorics.PerfectMatching.Model
-import OAI.Combinatorics.PerfectMatching.ObjectivePenalty
-import OAI.Combinatorics.PerfectMatching.ParameterBounds
-import OAI.Combinatorics.PerfectMatching.Partition
-import OAI.Combinatorics.PerfectMatching.Separable
-import OAI.Combinatorics.PerfectMatching.SeparableCount
-import OAI.Combinatorics.PerfectMatching.SeparableCovariance
-import OAI.Combinatorics.PerfectMatching.SeparableLower
-import OAI.Combinatorics.PerfectMatching.SizeBounds
-import OAI.Combinatorics.PerfectMatching.Violation
-import OAI.Combinatorics.PerfectMatching.WeightedSpectrum
-import OAI.Combinatorics.PlanarL1.Embedding
-import OAI.Combinatorics.ProgressionColoring.Main
-import OAI.Combinatorics.Ryser.Main
-import OAI.Combinatorics.Ryser.OddExtensions
-import OAI.Combinatorics.SecondNeighborhood.Main
-import OAI.Combinatorics.Sensitivity.Main
-import OAI.Combinatorics.SharpRamsey.Main
-import OAI.Combinatorics.SharpThreshold.Main
-import OAI.Combinatorics.Sidorenko.RandomHost
-import OAI.Combinatorics.SnakyCertificate.Main
-import OAI.Combinatorics.SparsestCut.Main
-import OAI.Combinatorics.SphericalRamsey
-import OAI.Combinatorics.SphericalRamsey.Result
-import OAI.Combinatorics.SquareDifference.Main
-import OAI.Combinatorics.ThinTrees.Main
-import OAI.Combinatorics.TreeEdit.Main
-import OAI.Combinatorics.TreewidthL1.Main
-import OAI.Combinatorics.TriangleRemoval.Main
-import OAI.Combinatorics.TwoWayAutomata.ExplicitFamily
-import OAI.Combinatorics.TwoWayAutomata.Main
-import OAI.Combinatorics.TwoWayAutomata.PolynomialBound
-import OAI.Combinatorics.UniformKServer.Main
-import OAI.Combinatorics.VariableWL
-import OAI.Combinatorics.YoungDiagram.RealCapacity
-import OAI.Combinatorics.YoungDiagram.RealLargeCapacity
-import OAI.Computability.BinPacking.Main
-import OAI.Computability.DegreeRigidity.Main
-import OAI.Computability.DepthThree
-import OAI.Computability.DeterministicSum.Main
-import OAI.Computability.DirectedFeedback.Main
-import OAI.Computability.FourierCircuit.Main
-import OAI.Computability.FourierTransform.Main
-import OAI.Computability.GeneralizedCircuits.Mesh
-import OAI.Computability.Logspace.Deterministic
-import OAI.Computability.Logspace.Equality
-import OAI.Computability.LoopMatching.Endpoint
-import OAI.Computability.MatchingCount.ArgRoutine
-import OAI.Computability.MatchingCount.BinaryAlgorithm
-import OAI.Computability.MatchingCount.BinaryParser
-import OAI.Computability.MatchingCount.BinarySolve
-import OAI.Computability.MatchingCount.CappedPower
-import OAI.Computability.MatchingCount.CutAlgebra
-import OAI.Computability.MatchingCount.DenseOracle
-import OAI.Computability.MatchingCount.DivisionRoutine
-import OAI.Computability.MatchingCount.FiniteStack
-import OAI.Computability.MatchingCount.FoldRoutine
-import OAI.Computability.MatchingCount.GuardRoutine
-import OAI.Computability.MatchingCount.IntegerRoutine
-import OAI.Computability.MatchingCount.IterateRoutine
-import OAI.Computability.MatchingCount.LengthRoutine
-import OAI.Computability.MatchingCount.ListNumeric
-import OAI.Computability.MatchingCount.ListRoutine
-import OAI.Computability.MatchingCount.MapRoutine
-import OAI.Computability.MatchingCount.OddCutParity
-import OAI.Computability.MatchingCount.OddCutUncross
-import OAI.Computability.MatchingCount.OddMincut
-import OAI.Computability.MatchingCount.OptionCases
-import OAI.Computability.MatchingCount.OracleBounds
-import OAI.Computability.MatchingCount.OrbitRoutine
-import OAI.Computability.MatchingCount.PairRoutine
-import OAI.Computability.MatchingCount.ParamRoutine
-import OAI.Computability.MatchingCount.ParserSupport
-import OAI.Computability.MatchingCount.PolyCoding
-import OAI.Computability.MatchingCount.ProjectionRoutine
-import OAI.Computability.MatchingCount.RawBitsBridge
-import OAI.Computability.MatchingCount.RawContraction
-import OAI.Computability.MatchingCount.RawContractionCompiler
-import OAI.Computability.MatchingCount.RawData
-import OAI.Computability.MatchingCount.RawFlow
-import OAI.Computability.MatchingCount.RawFlowCompiler
-import OAI.Computability.MatchingCount.RawFlowSize
-import OAI.Computability.MatchingCount.RawInput
-import OAI.Computability.MatchingCount.RawOddBounds
-import OAI.Computability.MatchingCount.RawOddCompiler
-import OAI.Computability.MatchingCount.RawOddCut
-import OAI.Computability.MatchingCount.RawOptimizer
-import OAI.Computability.MatchingCount.RawOptimizerCompiler
-import OAI.Computability.MatchingCount.RawOptimizerData
-import OAI.Computability.MatchingCount.RawOptimizerDataCompiler
-import OAI.Computability.MatchingCount.RawOptimizerRows
-import OAI.Computability.MatchingCount.RawOptimizerRowsCompiler
-import OAI.Computability.MatchingCount.RawPrimitives
-import OAI.Computability.MatchingCount.RawRoutine
-import OAI.Computability.MatchingCount.RawSearch
-import OAI.Computability.MatchingCount.RawSearchCompiler
-import OAI.Computability.MatchingCount.RawSearchSize
-import OAI.Computability.MatchingCount.RawSize
-import OAI.Computability.MatchingCount.RecursiveBounds
-import OAI.Computability.MatchingCount.RecursiveCoding
-import OAI.Computability.MatchingCount.RecursiveCompiler
-import OAI.Computability.MatchingCount.RecursiveEval
-import OAI.Computability.MatchingCount.ResidualSearch
-import OAI.Computability.MatchingCount.ScaledAlgorithm
-import OAI.Computability.MatchingCount.ScaledBounds
-import OAI.Computability.MatchingCount.ScaledState
-import OAI.Computability.MatchingCount.SelectRoutine
-import OAI.Computability.MatchingCount.SequenceRoutine
-import OAI.Computability.MatchingCount.StackCopy
-import OAI.Computability.MatchingCount.StackEmbed
-import OAI.Computability.MatchingCount.StackFrame
-import OAI.Computability.MatchingCount.StackMacros
-import OAI.Computability.MatchingCount.StackPair
-import OAI.Computability.MatchingCount.StackRoutine
-import OAI.Computability.MatchingCount.SumRoutine
-import OAI.Computability.MatchingCount.TerminalMincut
-import OAI.Computability.MatchingCount.UnaryArithmetic
-import OAI.Computability.MatchingCount.UnaryClog
-import OAI.Computability.MatchingCount.UnaryDivision
-import OAI.Computability.MatchingCount.UnitFlow
-import OAI.Computability.MatchingCount.UnitMincut
-import OAI.Computability.MaxCut.Main
-import OAI.Computability.MeanPayoff.Main
-import OAI.Computability.MinUncut.Main
-import OAI.Computability.PerfectCompleteness.Main
-import OAI.Computability.QuantumFactoring.Main
-import OAI.Computability.RandomMean.Main
-import OAI.Computability.RationalHitting.Main
-import OAI.Computability.Scheduling.Main
-import OAI.Computability.SpaceSimulation.Main
-import OAI.Computability.StarHeight.Main
-import OAI.Computability.Superstring.Main
-import OAI.Computability.TypeSystem.Normalization
-import OAI.Computability.UniqueGames.Main
-import OAI.Computability.VertexCover.Main
-import OAI.Computability.WLIdentification.Main
-import OAI.Computability.WeisfeilerLeman.Main
-import OAI.Computability.WitnessedChoice.Main
-import OAI.Dynamics.ConditionalShuffle.TwoColors
-import OAI.Dynamics.EntropyCounterexample
-import OAI.Dynamics.MultipleMixing.Main
-import OAI.Dynamics.SmoothObstruction.Main
-import OAI.Dynamics.StandardMap.ComponentsMain
-import OAI.Dynamics.StandardMap.EntropyEndpoint
-import OAI.Dynamics.StandardMap.EntropyLyapunov
-import OAI.Dynamics.ThreeTorus.Main
-import OAI.Dynamics.TriangleBilliards.Main
-import OAI.GameTheory.SnakyConditional.Main
-import OAI.GameTheory.SnakyTwentyOne.Main
-import OAI.Geometry.Anticanonical.BranchedQuotient
-import OAI.Geometry.Anticanonical.DeckRecovery
-import OAI.Geometry.Anticanonical.DeletedFaces
-import OAI.Geometry.Anticanonical.HilbertEstimates
-import OAI.Geometry.Arnold.Main
-import OAI.Geometry.BallPacking.Main
-import OAI.Geometry.BallPacking.Necessity.Main
-import OAI.Geometry.Borsuk.Main
-import OAI.Geometry.CAT0Fillings.Main
-import OAI.Geometry.Cannon.Main
-import OAI.Geometry.ConjugatePoints.ThreeManifold
-import OAI.Geometry.CoveringDensity.Main
-import OAI.Geometry.CubicTorus.Main
-import OAI.Geometry.CylinderCovering.Main
-import OAI.Geometry.DoublingHilbert.CompactBanach
-import OAI.Geometry.DoublingHilbert.Main
-import OAI.Geometry.HarmonicGrowth.Main
-import OAI.Geometry.HeilbronnTriangle
-import OAI.Geometry.HyperbolicGroups.BipartiteLink
-import OAI.Geometry.HyperbolicGroups.ClosedPicture
-import OAI.Geometry.HyperbolicGroups.LetterFillingArea
-import OAI.Geometry.HyperbolicGroups.LocalSegmentContinuation
-import OAI.Geometry.HyperbolicGroups.LocalSegmentTools
-import OAI.Geometry.HyperbolicGroups.Main
-import OAI.Geometry.HyperbolicGroups.Parity
-import OAI.Geometry.HyperbolicGroups.PolygonContraction
-import OAI.Geometry.HyperbolicGroups.TreePathDepth
-import OAI.Geometry.HypersurfaceGerms.Main
-import OAI.Geometry.Immersion.ClosedSurface.Main
-import OAI.Geometry.IntegralFillings.Main
-import OAI.Geometry.IsometricImmersion.All
-import OAI.Geometry.Kahler
-import OAI.Geometry.KahlerSplitting.Main
-import OAI.Geometry.LatticeCovering
-import OAI.Geometry.LogVolume.BrunnMinkowski
-import OAI.Geometry.Manifold.ProjectiveSphere
-import OAI.Geometry.NearbyLagrangian.Main
-import OAI.Geometry.NearbyLagrangian.TubeKernel
-import OAI.Geometry.NegativeCurvature.Tube
-import OAI.Geometry.NilpotentCharts.Main
-import OAI.Geometry.NodalSets.Main
-import OAI.Geometry.Perimeter.Regularity.Main
-import OAI.Geometry.PeriodicTiling.Main
-import OAI.Geometry.PinnedDistances.Pins
-import OAI.Geometry.PlaneColoring.Five
-import OAI.Geometry.PlaneColoring.Main
-import OAI.Geometry.PolarProducts.Main
-import OAI.Geometry.PrescribedPotential.AtlasLocalization
-import OAI.Geometry.PrescribedPotential.AugmentedVolume
-import OAI.Geometry.PrescribedPotential.CalabiFrameCalculus
-import OAI.Geometry.PrescribedPotential.CalabiFrameScalars
-import OAI.Geometry.PrescribedPotential.CalabiRicciContraction
-import OAI.Geometry.PrescribedPotential.ClosedRangeSmooth
-import OAI.Geometry.PrescribedPotential.CommutatorSupport
-import OAI.Geometry.PrescribedPotential.CompletedProduct
-import OAI.Geometry.PrescribedPotential.CompletedRealification
-import OAI.Geometry.PrescribedPotential.CompletedVolume
-import OAI.Geometry.PrescribedPotential.CoordinateEllipticity
-import OAI.Geometry.PrescribedPotential.CoordinateExtension
-import OAI.Geometry.PrescribedPotential.CoordinateMinimumBound
-import OAI.Geometry.PrescribedPotential.CoordinatePotential
-import OAI.Geometry.PrescribedPotential.DerivativeEllipticGain
-import OAI.Geometry.PrescribedPotential.DerivativeLocalization
-import OAI.Geometry.PrescribedPotential.EvenRellich
-import OAI.Geometry.PrescribedPotential.FixedScaleBootstrap
-import OAI.Geometry.PrescribedPotential.FredholmKernel
-import OAI.Geometry.PrescribedPotential.FredholmRegularity
-import OAI.Geometry.PrescribedPotential.GlobalBounds
-import OAI.Geometry.PrescribedPotential.GlobalError
-import OAI.Geometry.PrescribedPotential.GlobalFiniteEmbedding
-import OAI.Geometry.PrescribedPotential.GlobalNonlinearCommutator
-import OAI.Geometry.PrescribedPotential.GlobalProduct
-import OAI.Geometry.PrescribedPotential.GlobalRellich
-import OAI.Geometry.PrescribedPotential.GlobalSmooth
-import OAI.Geometry.PrescribedPotential.HigherMetricOperator
-import OAI.Geometry.PrescribedPotential.HigherPoisson
-import OAI.Geometry.PrescribedPotential.HigherResolvent
-import OAI.Geometry.PrescribedPotential.LinearizedCommutator
-import OAI.Geometry.PrescribedPotential.MetricQuasilinearEquation
-import OAI.Geometry.PrescribedPotential.NonlinearCoreCommutator
-import OAI.Geometry.PrescribedPotential.NonlinearRemainderStrong
-import OAI.Geometry.PrescribedPotential.PathCalabiBound
-import OAI.Geometry.PrescribedPotential.PathCoordinateDensity
-import OAI.Geometry.PrescribedPotential.PathFrameBounds
-import OAI.Geometry.PrescribedPotential.PointPoisson
-import OAI.Geometry.PrescribedPotential.PositiveOpenness
-import OAI.Geometry.PrescribedPotential.PositivePotentialLinearization
-import OAI.Geometry.PrescribedPotential.PotentialDensityDifferential
-import OAI.Geometry.PrescribedPotential.PrescribedPotential
-import OAI.Geometry.PrescribedPotential.QuantitativeResolvent
-import OAI.Geometry.PrescribedPotential.RealMapSmooth
-import OAI.Geometry.PrescribedPotential.RealPoisson
-import OAI.Geometry.PrescribedPotential.RealSmoothCompactness
-import OAI.Geometry.PrescribedPotential.RegularityCutoffs
-import OAI.Geometry.PrescribedPotential.RegularityJets
-import OAI.Geometry.PrescribedPotential.SmoothChartInverse
-import OAI.Geometry.PrescribedPotential.SmoothLocalPath
-import OAI.Geometry.PrescribedPotential.SmoothPathLimit
-import OAI.Geometry.PrescribedPotential.SmoothSequenceCompactness
-import OAI.Geometry.PrescribedPotential.SobolevDetDifferential
-import OAI.Geometry.PrescribedPotential.SobolevExponential
-import OAI.Geometry.PrescribedPotential.SobolevFirstGain
-import OAI.Geometry.PrescribedPotential.SobolevOpenness
-import OAI.Geometry.PrescribedPotential.TwoScaleRegularity
-import OAI.Geometry.PrescribedPotential.UniformPathSobolev
-import OAI.Geometry.PrescribedPotential.WeakDerivativeLocalization
-import OAI.Geometry.PrescribedPotential.WeakPositive
-import OAI.Geometry.PrescribedRicci.BoundedSmoothPoisson
-import OAI.Geometry.PrescribedRicci.CalabiGradient
-import OAI.Geometry.PrescribedRicci.CalabiMetricC1
-import OAI.Geometry.PrescribedRicci.CalabiRough
-import OAI.Geometry.PrescribedRicci.CalabiTrace
-import OAI.Geometry.PrescribedRicci.CalabiTraceBound
-import OAI.Geometry.PrescribedRicci.ChernBianchi
-import OAI.Geometry.PrescribedRicci.ChernLuConnection
-import OAI.Geometry.PrescribedRicci.ChernRough
-import OAI.Geometry.PrescribedRicci.CofactorCore
-import OAI.Geometry.PrescribedRicci.CofactorCoreEquation
-import OAI.Geometry.PrescribedRicci.CofactorLocalization
-import OAI.Geometry.PrescribedRicci.CompactPathBounds
-import OAI.Geometry.PrescribedRicci.CompactPathRicci
-import OAI.Geometry.PrescribedRicci.FixedCofactorBootstrap
-import OAI.Geometry.PrescribedRicci.FixedCofactorStep
-import OAI.Geometry.PrescribedRicci.GlobalSmoothCompact
-import OAI.Geometry.PrescribedRicci.IntegratedKahlerEnergy
-import OAI.Geometry.PrescribedRicci.JetProductL2
-import OAI.Geometry.PrescribedRicci.KahlerCauchySchwarz
-import OAI.Geometry.PrescribedRicci.KahlerChartGradient
-import OAI.Geometry.PrescribedRicci.KahlerChartLp
-import OAI.Geometry.PrescribedRicci.KahlerEnergyProduct
-import OAI.Geometry.PrescribedRicci.KahlerIntegralPos
-import OAI.Geometry.PrescribedRicci.KahlerL2Comparison
-import OAI.Geometry.PrescribedRicci.KahlerPoincare
-import OAI.Geometry.PrescribedRicci.KahlerPoissonL2
-import OAI.Geometry.PrescribedRicci.KahlerSobolev
-import OAI.Geometry.PrescribedRicci.KahlerVolumeMeasure
-import OAI.Geometry.PrescribedRicci.KahlerVolumeVariation
-import OAI.Geometry.PrescribedRicci.MongeAmpereC0
-import OAI.Geometry.PrescribedRicci.MongeAmpereEnergy
-import OAI.Geometry.PrescribedRicci.MongeAmpereL2
-import OAI.Geometry.PrescribedRicci.NormalizeVolumePath
-import OAI.Geometry.PrescribedRicci.PathLocalGain
-import OAI.Geometry.PrescribedRicci.PathUniformHigher
-import OAI.Geometry.PrescribedRicci.PathUniformSobolevBase
-import OAI.Geometry.PrescribedRicci.PositiveOpenness
-import OAI.Geometry.PrescribedRicci.PositivePathLimit
-import OAI.Geometry.PrescribedRicci.PrescribedRicci
-import OAI.Geometry.PrescribedRicci.SmoothLocalInverse
-import OAI.Geometry.PrescribedRicci.SmoothPathLimit
-import OAI.Geometry.PrescribedRicci.SmoothPositivePath
-import OAI.Geometry.PrescribedRicci.TameCofactorGain
-import OAI.Geometry.PrescribedRicci.TraceMaximum
-import OAI.Geometry.PrescribedRicci.UniformCalabi
-import OAI.Geometry.PrescribedRicci.UniformChernLu
-import OAI.Geometry.PrescribedRicci.UniformMetricBounds
-import OAI.Geometry.PrescribedRicci.UniformPathPatches
-import OAI.Geometry.PrescribedRicci.WeightedMongeAmpereEnergy
-import OAI.Geometry.ProjectionBodies.Main
-import OAI.Geometry.ProjectionBody.Main
-import OAI.Geometry.ProjectionVolume.Main
-import OAI.Geometry.QuadricBundles.Main
-import OAI.Geometry.Riemannian.HarmonicCore.Main
-import OAI.Geometry.RuledCovering.Main
-import OAI.Geometry.SmoothYau.Main
-import OAI.Geometry.SplitTangent.AllCompactLeaves
-import OAI.Geometry.SplitTangent.AllProjectiveLeaves
-import OAI.Geometry.SplitTangent.AnalyticFiniteRelations
-import OAI.Geometry.SplitTangent.ArbitraryPlaqueLeaves
-import OAI.Geometry.SplitTangent.BasicDerivativeKernel
-import OAI.Geometry.SplitTangent.BoxFirstTransport
-import OAI.Geometry.SplitTangent.ChartSubmersion
-import OAI.Geometry.SplitTangent.CompactLeafQuotient
-import OAI.Geometry.SplitTangent.CompactPositiveLeaf
-import OAI.Geometry.SplitTangent.ComplementaryLeafIntersections
-import OAI.Geometry.SplitTangent.DenseCompactLeaves
-import OAI.Geometry.SplitTangent.DenseRationalVariation
-import OAI.Geometry.SplitTangent.EmbeddedLeafCharts
-import OAI.Geometry.SplitTangent.EmbeddedRationalCurve
-import OAI.Geometry.SplitTangent.EmbeddedTangentLift
-import OAI.Geometry.SplitTangent.EverywhereSmoothLeafQuotient
-import OAI.Geometry.SplitTangent.GenericLeafProductCover
-import OAI.Geometry.SplitTangent.GlobalAffineRelations
-import OAI.Geometry.SplitTangent.GlobalLeafRelations
-import OAI.Geometry.SplitTangent.GlobalProducts
-import OAI.Geometry.SplitTangent.GlobalTopologicalProduct
-import OAI.Geometry.SplitTangent.HolomorphicBasicTransport
-import OAI.Geometry.SplitTangent.HolomorphicLeafBasic
-import OAI.Geometry.SplitTangent.HolomorphicManifoldProduct
-import OAI.Geometry.SplitTangent.IntegrabilityReduction
-import OAI.Geometry.SplitTangent.IntrinsicPositiveLeaf
-import OAI.Geometry.SplitTangent.LeafAffineRelations
-import OAI.Geometry.SplitTangent.LeafAlgebraicDependence
-import OAI.Geometry.SplitTangent.LeafClosureRelations
-import OAI.Geometry.SplitTangent.LeafCompactLimits
-import OAI.Geometry.SplitTangent.LeafGermIdentity
-import OAI.Geometry.SplitTangent.LeafHomogeneousRelations
-import OAI.Geometry.SplitTangent.LeafLimitRelations
-import OAI.Geometry.SplitTangent.LeafLimitsSinglePlaque
-import OAI.Geometry.SplitTangent.LeafPolynomialGrowth
-import OAI.Geometry.SplitTangent.LeafProducts
-import OAI.Geometry.SplitTangent.LeafTopology
-import OAI.Geometry.SplitTangent.ManifoldJetAnalytic
-import OAI.Geometry.SplitTangent.OpenPositiveLeaves
-import OAI.Geometry.SplitTangent.PlaqueConnectedSets
-import OAI.Geometry.SplitTangent.PlaqueTangentImage
-import OAI.Geometry.SplitTangent.PolynomialLeafGerms
-import OAI.Geometry.SplitTangent.PositiveCoverLift
-import OAI.Geometry.SplitTangent.ProductCriteria
-import OAI.Geometry.SplitTangent.ProjectiveHypersurfaces
-import OAI.Geometry.SplitTangent.ProjectivePositiveLeaf
-import OAI.Geometry.SplitTangent.ProjectiveTangentBox
-import OAI.Geometry.SplitTangent.QuotientSubmersions
-import OAI.Geometry.SplitTangent.RCCovariantVanishing
-import OAI.Geometry.SplitTangent.RationalCoverLift
-import OAI.Geometry.SplitTangent.RationalReparam
-import OAI.Geometry.SplitTangent.RationalVariation
-import OAI.Geometry.SplitTangent.TangentCurveLeaf
-import OAI.Geometry.SplitTangent.TangentRationalChains
-import OAI.Geometry.SplitTangent.TransverseDiscrete
-import OAI.Geometry.SplitTangent.TransverseQuotient
-import OAI.Geometry.SplitTangent.ZariskiPairOpen
-import OAI.Geometry.Symplectic.MorseBounds.Main
-import OAI.Geometry.TamingCompatibility.Main
-import OAI.Geometry.TranslativeCovering.Main
-import OAI.Geometry.TriangularCovering.Main
-import OAI.Geometry.UnitDistances.Main
-import OAI.Geometry.Varifold.NullComparison
-import OAI.Geometry.WeakHessian.Main
-import OAI.Geometry.WeakMTW.Main
-import OAI.GroupTheory.ArithmeticGerms.Stabilizers
-import OAI.GroupTheory.ArtinCAT0.Main
-import OAI.GroupTheory.BooneHigman.Main
-import OAI.GroupTheory.CyclicSeries.Length
-import OAI.GroupTheory.FiniteIndex.Realization
-import OAI.GroupTheory.FiniteKernel.VirtualAbelian
-import OAI.GroupTheory.FiniteLattice.LabelledCollection
-import OAI.GroupTheory.FiniteLattice.NormalPreimage
-import OAI.GroupTheory.FiniteType.Main
-import OAI.GroupTheory.Heisenberg.BarDetector
-import OAI.GroupTheory.Hyperbolic.Main
-import OAI.GroupTheory.Kervaire.Main
-import OAI.GroupTheory.PChains.Cancellation
-import OAI.GroupTheory.PeriodicGroups.Main
-import OAI.GroupTheory.SimpleAmenable.Main
-import OAI.GroupTheory.SimpleOvergroups.Main
-import OAI.GroupTheory.Thompson.Main
-import OAI.GroupTheory.UniversalGroup.Main
-import OAI.InformationTheory.AmplitudeDamping.Main
-import OAI.InformationTheory.BooleanNoise.Main
-import OAI.InformationTheory.DimensionTen.StateKey
-import OAI.InformationTheory.Entanglement.Main
-import OAI.InformationTheory.PhotonNumber.Inequality
-import OAI.InformationTheory.QuantumCircuit.Main
-import OAI.InformationTheory.QuantumCircuit.Regularity
-import OAI.InformationTheory.SecretKey.DimensionTen
-import OAI.InformationTheory.SecretKey.Main
-import OAI.InformationTheory.SoftChannel.Main
-import OAI.LinearAlgebra.Barker.Main
-import OAI.LinearAlgebra.CirculantHadamard.Main
-import OAI.LinearAlgebra.CliffordAlgebra.KugaSatake
-import OAI.LinearAlgebra.Descent.Rational
-import OAI.LinearAlgebra.Hadamard.Main
-import OAI.LinearAlgebra.MatrixFields.Main
-import OAI.LinearAlgebra.MatrixMultiplication.Main
-import OAI.LinearAlgebra.TensorProduct.ScalarDescent
-import OAI.MathematicalPhysics.AlternatingFlow.Development
-import OAI.MathematicalPhysics.Anderson.Certificates
-import OAI.MathematicalPhysics.BFSS.Main
-import OAI.MathematicalPhysics.Coulomb.OuterRadius
-import OAI.MathematicalPhysics.CriticalMixing.Main
-import OAI.MathematicalPhysics.CriticalSK.LowerExponents
-import OAI.MathematicalPhysics.CriticalSK.Main
-import OAI.MathematicalPhysics.Elasticity.Uniqueness
-import OAI.MathematicalPhysics.HardSphere.GrandCanonical
-import OAI.MathematicalPhysics.Heisenberg.Main
-import OAI.MathematicalPhysics.NavierStokes.BalancedTransport.MainResult
-import OAI.MathematicalPhysics.NavierStokes.Material.Main
-import OAI.MathematicalPhysics.NavierStokes.ShearFlows.Main
-import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.Main
-import OAI.MathematicalPhysics.PEPSFilters.GeometricOptimizer
-import OAI.MathematicalPhysics.PEPSMove.PhysicalMove
-import OAI.MathematicalPhysics.PEPSSubvolume.Main
-import OAI.MathematicalPhysics.PlanarAnderson.Spectrum
-import OAI.MathematicalPhysics.PrefixFlows.Main
-import OAI.MathematicalPhysics.RapidForcing.Main
-import OAI.MathematicalPhysics.RecorderFlows.Main
-import OAI.MathematicalPhysics.SheetFlows.SheetProgram
-import OAI.MathematicalPhysics.TensorNetwork.VectorColumn
-import OAI.MeasureTheory.DyadicAvoidance.Main
-import OAI.MeasureTheory.Falconer.Main
-import OAI.MeasureTheory.FalconerCompletion.Main
-import OAI.MeasureTheory.SelfSimilar.Main
-import OAI.MeasureTheory.SelfSimilar.DimensionCorollaries
-import OAI.ModelTheory.AbstractElementary.ModelExistence
-import OAI.ModelTheory.Categoricity.Main
-import OAI.ModelTheory.Choiceless.Separation
-import OAI.NumberTheory.Catalan.Main
-import OAI.NumberTheory.CubicGauss.Main
-import OAI.NumberTheory.CubicGram.Main
-import OAI.NumberTheory.DirichletL.Hecke.Nonvanishing
-import OAI.NumberTheory.DirichletL.Nonvanishing
-import OAI.NumberTheory.DuffinSchaeffer.Reductions
-import OAI.NumberTheory.EgyptianFractions.All
-import OAI.NumberTheory.GaussianMoat.Main
-import OAI.NumberTheory.Jacobsthal.Main
-import OAI.NumberTheory.JointDickman.PaperMain
-import OAI.NumberTheory.NumberField.UnitRadicals
-import OAI.NumberTheory.OrdinaryCorrelations.AbsoluteDefect.Main
-import OAI.NumberTheory.OrdinaryCorrelations.Elliott.Main
-import OAI.NumberTheory.OrdinaryCorrelations.HighTrace.Main
-import OAI.NumberTheory.Ostmann.Main
-import OAI.NumberTheory.PiExponent.Main
-import OAI.NumberTheory.PowerFree.Main
-import OAI.NumberTheory.PrimeGaps.RatioCorollary
-import OAI.NumberTheory.PrimePackets.Main
-import OAI.NumberTheory.QuadraticForms.ComponentLengths
-import OAI.NumberTheory.ShortEgyptian.Main
-import OAI.NumberTheory.SiegelZeros.Main
-import OAI.NumberTheory.SingleFold.Main
-import OAI.NumberTheory.TotientAsymptotic.Main
-import OAI.NumberTheory.TotientFibers.FiniteFiber
-import OAI.Optimization.Tropical.TruffetCounterexample
-import OAI.Probability.Ballisticity.Main
-import OAI.Probability.Ballisticity.VelocityHemisphere
-import OAI.Probability.BenjaminiSchramm
-import OAI.Probability.BinarySweep.Main
-import OAI.Probability.ClassicalON.Main
-import OAI.Probability.Contingency.LocalRepair
-import OAI.Probability.Contingency.PreparedResidual
-import OAI.Probability.ContinuumTransition.Main
-import OAI.Probability.CoordinateSweeps.Main
-import OAI.Probability.CriticalPercolation.Main
-import OAI.Probability.CriticalZ3.Main
-import OAI.Probability.CubeShuffle.WeightedMoments
-import OAI.Probability.DilutedSpin.Main
-import OAI.Probability.DirectionalWalk.Main
-import OAI.Probability.EntangledGames.Main
-import OAI.Probability.FKPressure.Main
-import OAI.Probability.FirstPassage.Main
-import OAI.Probability.FKMaps.Convergence
-import OAI.Probability.FreeIsing.Main
-import OAI.Probability.GaussianInformation.Main
-import OAI.Probability.GaussianPropeller.Main
-import OAI.Probability.GaussianRegression.Main
-import OAI.Probability.GaussianReplacement.Streaming
-import OAI.Probability.Honeycomb.FreeEnergy
-import OAI.Probability.Honeycomb.TriangleVisits
-import OAI.Probability.Ising.Main
-import OAI.Probability.IsingPerceptron.FullReplicaIBP
-import OAI.Probability.IsingPerceptron.VariationalFiniteness
-import OAI.Probability.JammingConcavity.DeletionUniformRowEstimates
-import OAI.Probability.JammingConcavity.RowFiniteDiagonalRowGeneralInterpolation
-import OAI.Probability.JammingConcavity.RowQuadraticTerminalLimit
-import OAI.Probability.JammingConcavity.VarianceTensorization
-import OAI.Probability.LatticeSurface.Brownian.BrownianClockFacet
-import OAI.Probability.LatticeSurface.Brownian.BrownianClockSupport
-import OAI.Probability.LatticeSurface.Brownian.BrownianPath
-import OAI.Probability.LatticeSurface.Brownian.BrownianPathIdentities
-import OAI.Probability.LatticeSurface.Coercivity.HoleWeighted
-import OAI.Probability.LatticeSurface.Convexity.ClusterDynamicsDefs
-import OAI.Probability.LatticeSurface.Convexity.FiniteLevelBarrier
-import OAI.Probability.LatticeSurface.Convexity.Metropolis
-import OAI.Probability.LatticeSurface.Convexity.MetropolisSupport
-import OAI.Probability.LatticeSurface.Convexity.ProductSums
-import OAI.Probability.LatticeSurface.Convexity.SupportUnit
-import OAI.Probability.LatticeSurface.Convexity.TailNumerics
-import OAI.Probability.LatticeSurface.Convexity.TransportResampling
-import OAI.Probability.LatticeSurface.Dynamics.CompactCoupling
-import OAI.Probability.LatticeSurface.Dynamics.LatticeShift
-import OAI.Probability.LatticeSurface.Dynamics.ProjectionComparison
-import OAI.Probability.LatticeSurface.Dynamics.ProjectionGlobal
-import OAI.Probability.LatticeSurface.Dynamics.ProjectionLimit
-import OAI.Probability.LatticeSurface.Dynamics.TiltApplication
-import OAI.Probability.LatticeSurface.Dynamics.TiltSupport
-import OAI.Probability.LatticeSurface.Dynamics.WeightedInitial
-import OAI.Probability.LatticeSurface.Geometry.ContactVolume
-import OAI.Probability.LatticeSurface.Moments.Averages
-import OAI.Probability.Localization.Main
-import OAI.Probability.LogConcave.Main
-import OAI.Probability.MatroidProphet
-import OAI.Probability.MatroidSecretary.Main
-import OAI.Probability.MemoryPrecision.Main
-import OAI.Probability.ParisiFinite.ParisiCanonicalTransport
-import OAI.Probability.ParisiFinite.ParisiEntropyDecomposition
-import OAI.Probability.ParisiFinite.RealShiftNormSqLe
-import OAI.Probability.ParisiFinite.SpinFlip
-import OAI.Probability.ParisiFinite.WeightedGradientField
-import OAI.Probability.Perceptron.Main
-import OAI.Probability.PosteriorReplicas.Main
-import OAI.Probability.ProbabilityMeasure.ArctanCenter
-import OAI.Probability.ProjectionMoments.Main
-import OAI.Probability.RadialTransition.DensityInterval
-import OAI.Probability.RadialTransition.Main
-import OAI.Probability.RandomSAT.Main
-import OAI.Probability.Renewal.Main
-import OAI.Probability.SATComputability.Main
-import OAI.Probability.SATVariance.Main
-import OAI.Probability.SKBarriers.Main
-import OAI.Probability.SKCutoff.Denominator
-import OAI.Probability.SKFluctuations.ComparisonMinimumRealOverlapConclusion
-import OAI.Probability.SKGap.Main
-import OAI.Probability.SKSupport.Main
-import OAI.Probability.SKValue.Main
-import OAI.Probability.SLE.MainReduction
-import OAI.Probability.SLE.MomentGauge
-import OAI.Probability.SLE.SausageCharge
-import OAI.Probability.SLEGauge.Main
-import OAI.Probability.SLELocalization.CircleMeans
-import OAI.Probability.SLELocalization.DriverStability
-import OAI.Probability.SLELocalization.FilledObstacle
-import OAI.Probability.SLELocalization.LoewnerCocycle
-import OAI.Probability.SLELocalization.MappedExit
-import OAI.Probability.SLELocalization.MeshSlab
-import OAI.Probability.SLELocalization.PastObstacle
-import OAI.Probability.SLELocalization.RetainedFirstMoment
-import OAI.Probability.SLELocalization.SlitMaps
-import OAI.Probability.SLELocalization.SpatialLimit
-import OAI.Probability.SLELocalization.StrongRestart
-import OAI.Probability.SLELocalization.TraceAvoidance
-import OAI.Probability.SLELocalization.TruncatedEnergy
-import OAI.Probability.SignedSweeps.Main
-import OAI.Probability.SpanningForest.Main
-import OAI.Probability.SphericalField.Main
-import OAI.Probability.SphericalPerceptron.PressureComparison
-import OAI.Probability.StrongRayleigh
-import OAI.Probability.Subsphere.Main
-import OAI.Probability.SwitchChain
-import OAI.Probability.ThorpCompatibility.Main
-import OAI.Probability.ThorpResults.FullDevelopment
-import OAI.Probability.ThorpRouting.Main
-import OAI.Probability.ThorpRouting.RegularTrace
-import OAI.Probability.ThorpShuffle.FirstFourier
-import OAI.Probability.ThorpShuffle.FirstMain
-import OAI.Probability.ThorpShuffle.FixedLists.Main
-import OAI.Probability.ThorpShuffle.Main
-import OAI.Probability.ThreeState.Main
-import OAI.Probability.ThreeStateClauses.Main
-import OAI.Probability.TraceReconstruction
-import OAI.Probability.UnimodularPercolation.CayleyMain
-import OAI.Probability.UnimodularPercolation.Main
-import OAI.Probability.UnimodularPercolation.UnimodularMain
-import OAI.Probability.Voronoi.CardyFormula
-import OAI.RepresentationTheory.AlperinWeights.Brauer
-import OAI.RepresentationTheory.Block.Defect
-import OAI.RepresentationTheory.FoulkesHowe
-import OAI.RepresentationTheory.FoulkesSixth.Main
-import OAI.RepresentationTheory.GraphModels.NormalActions
-import OAI.RepresentationTheory.KazhdanLusztig.Invariance
-import OAI.RepresentationTheory.PartialPermutation.Main
-import OAI.RepresentationTheory.ProjectiveInduction.CosetTransport
-import OAI.RepresentationTheory.RestrictedTilting.Main
-import OAI.RepresentationTheory.RowColumn.Occupied
-import OAI.RepresentationTheory.Saxl.Main
-import OAI.RepresentationTheory.SpinAngle.TypeAngle
-import OAI.RepresentationTheory.TensorSquares.Main
-import OAI.RepresentationTheory.Torus.ConvexWeights
-import OAI.RepresentationTheory.VertexAlgebra.Main
-import OAI.RingTheory.BassTrace.Main
-import OAI.RingTheory.BassTrace.TorsionFree
-import OAI.RingTheory.DirectFiniteness.FinitelyPresented
-import OAI.RingTheory.DirectFiniteness.Main
-import OAI.RingTheory.DivisionRing.FiniteDescent
-import OAI.RingTheory.FormalGroup.Main
-import OAI.RingTheory.GroupAction.TraceOne
-import OAI.RingTheory.Kurosh.Main
-import OAI.RingTheory.Multiplicity.Main
-import OAI.RingTheory.Tachikawa.Counterexample
-import OAI.SetTheory.PartitionConsistency.Main
-import OAI.SetTheory.PartitionPrinciple.Main
-import OAI.Topology.ArtinGroups.Main
-import OAI.Topology.ArtinGroups.ParabolicIntersections.Main
-import OAI.Topology.EilenbergGanea.Main
-import OAI.Topology.TorusCompression.Boundary
-import OAI.Topology.CoarseAssembly
-import OAI.RepresentationTheory.UniversalSquare.Main
-import OAI.Probability.SKRatio.Main
-import OAI.Geometry.Relativity.CKS.Main
-import OAI.Geometry.SurfaceImmersion.All
-
-import OAI.GroupTheory.PolycyclicRecognition
+import OAI.Algebra.AffineCancellation.Main
+import OAI.Algebra.AlgebraicObstruction.AlgebraicInput
+import OAI.Algebra.AlgebraicObstruction.Holonomy
+import OAI.Algebra.AuslanderReiten.All
+import OAI.Algebra.BogomolovPop.Injectivity
+import OAI.Algebra.Cohomology.SemilinearCochains
+import OAI.Algebra.DepthFive.Main
+import OAI.Algebra.Determinantal.Main
+import OAI.Algebra.Divisor.Normalization
+import OAI.Algebra.Divisor.RationalRecovery
+import OAI.Algebra.Drinfeld.Completion
+import OAI.Algebra.FiniteTensor.Main
+import OAI.Algebra.Finitistic.Main
+import OAI.Algebra.FinitisticAsymmetry.Main
+import OAI.Algebra.FormalGroup.Honda.Main
+import OAI.Algebra.FormulaHitting.Main
+import OAI.Algebra.FrobeniusObstruction.Obstruction
+import OAI.Algebra.GroupRing.CylinderRetention
+import OAI.Algebra.GroupRing.Main
+import OAI.Algebra.MarkedTensor.Main
+import OAI.Algebra.Module.CompactPadic
+import OAI.Algebra.OddKaplansky.GeneralTransfer
+import OAI.Algebra.PolynomialChains.Real
+import OAI.Algebra.Universal.Coset
+import OAI.Algebra.Universal.GraphDecidability
+import OAI.AlgebraicGeometry.AbhyankarSathaye.Counterexample
+import OAI.AlgebraicGeometry.CartierSections
+import OAI.AlgebraicGeometry.CharacterVarieties
+import OAI.AlgebraicGeometry.CommutingDerivations.OrdinaryExtraMain
+import OAI.AlgebraicGeometry.LogKodaira.FiberNegative
+import OAI.AlgebraicGeometry.LogKodaira.RootFactor
+import OAI.AlgebraicGeometry.NumericalDimension.IndexedTransfer
+import OAI.AlgebraicGeometry.NumericalDimension.RelativeNegativity
+import OAI.AlgebraicGeometry.NumericalDimension.TwistedPushforward
+import OAI.AlgebraicGeometry.PlaneCurves.Nagata
+import OAI.AlgebraicGeometry.RelativeTriviality.RelativeTriviality
+import OAI.AlgebraicGeometry.SectionFields.Main
+import OAI.AlgebraicGeometry.Seshadri.Main
+import OAI.AlgebraicGeometry.Stability.NumericalWalls
+import OAI.AlgebraicGeometry.StableCoordinate.All
+import OAI.AlgebraicGeometry.SurfaceCones
+import OAI.Analysis.AffineBernstein.Main
+import OAI.Analysis.AllenCahn.Main
+import OAI.Analysis.AncientFlat.Main
+import OAI.Analysis.BackwardIntertwiners.AlgebraCorollary
+import OAI.Analysis.BernoulliCone.Main
+import OAI.Analysis.BiholderTransport.Main
+import OAI.Analysis.BoundedHochschild.AtomicPrimitives
+import OAI.Analysis.BoundedHochschild.CenterPowers
+import OAI.Analysis.BoundedHochschild.CentralLinearPatch
+import OAI.Analysis.BoundedHochschild.CocycleNormalParts
+import OAI.Analysis.BoundedHochschild.CommutativePrimitives
+import OAI.Analysis.BoundedHochschild.FactorFinitePaths
+import OAI.Analysis.BoundedHochschild.FactorPrimitive
+import OAI.Analysis.BoundedHochschild.FreeWords
+import OAI.Analysis.BoundedHochschild.MainResult
+import OAI.Analysis.BoundedHochschild.MatrixDisplacement
+import OAI.Analysis.BoundedHochschild.TypeIICorners
+import OAI.Analysis.BoundedHochschild.TypeIPrimitives
+import OAI.Analysis.BoundedHochschild.UnitaryLifts
+import OAI.Analysis.BoxTransport.Routing
+import OAI.Analysis.Brenier.Stability
+import OAI.Analysis.BrownMeasure.Main
+import OAI.Analysis.C0Absorption.Main
+import OAI.Analysis.CStarAlgebra.PureInfiniteness.Main
+import OAI.Analysis.CardinalInterpolation.ModTwelve
+import OAI.Analysis.CharacterCriterion.Main
+import OAI.Analysis.CircleDomains.LengthArea
+import OAI.Analysis.CircleDomains.Development
+import OAI.Analysis.Commutator.Main
+import OAI.Analysis.Conductivity.Main
+import OAI.Analysis.Cotype.Main
+import OAI.Analysis.CoulombIonization.Main
+import OAI.Analysis.CoulombRadii.Main
+import OAI.Analysis.CoulombTransport.Main
+import OAI.Analysis.Crouzeix.Main
+import OAI.Analysis.Daugavet.Main
+import OAI.Analysis.DiamondDistortion.Main
+import OAI.Analysis.DirectCrouzeix.Main
+import OAI.Analysis.DiskMaximal.Main
+import OAI.Analysis.FactorGeneration.Main
+import OAI.Analysis.FiniteFactor.Main
+import OAI.Analysis.ForestSpace.Main
+import OAI.Analysis.Fourier.HeightLayers
+import OAI.Analysis.Fourier.SphereExtension
+import OAI.Analysis.FourierExtension.LpEstimate
+import OAI.Analysis.FourierLLogL.PaperMain
+import OAI.Analysis.FreeEntropy.Main
+import OAI.Analysis.FreeGroupFactor.Main
+import OAI.Analysis.FreeWords.Main
+import OAI.Analysis.Graph.Gradients
+import OAI.Analysis.GroupDeterminants.Main
+import OAI.Analysis.HardSphere.Main
+import OAI.Analysis.HenonEmden.Main
+import OAI.Analysis.HilbertCrouzeix.Main
+import OAI.Analysis.HotSpots.Main
+import OAI.Analysis.HyperbolicCones
+import OAI.Analysis.Hyperinvariant
+import OAI.Analysis.IncompressibleFlow.AffineExtension
+import OAI.Analysis.InnerProductSpace.ComplexSolvability
+import OAI.Analysis.IntegralMeans.Main
+import OAI.Analysis.IntegralMeans.SharpEndpoints
+import OAI.Analysis.InterpolatedFactors.Interpolation
+import OAI.Analysis.IrrationalRotation.Main
+import OAI.Analysis.JiangSu.Main
+import OAI.Analysis.Kakeya.FixedScale
+import OAI.Analysis.Kaplansky.Main
+import OAI.Analysis.Kaplansky.StableFiniteness
+import OAI.Analysis.Laughlin.Main
+import OAI.Analysis.Laughlin.UniformGap
+import OAI.Analysis.LaughlinFock.Main
+import OAI.Analysis.LaughlinFock.TensorIntegrals
+import OAI.Analysis.LaughlinGap.Main
+import OAI.Analysis.LaughlinGap.ThreeAveraging
+import OAI.Analysis.LiebThirring.CoerciveInverse
+import OAI.Analysis.LiebThirring.CompactAlternative
+import OAI.Analysis.LiebThirring.ContinuousRepresentative
+import OAI.Analysis.LiebThirring.DiscreteSpectrum
+import OAI.Analysis.LiebThirring.Eigenspace
+import OAI.Analysis.LiebThirring.EigenvalueSum
+import OAI.Analysis.LiebThirring.FormCongruence
+import OAI.Analysis.LiebThirring.FreeForm
+import OAI.Analysis.LiebThirring.LocalCompactness
+import OAI.Analysis.LiebThirring.LocalWeight
+import OAI.Analysis.LiebThirring.LowestEigenvalue
+import OAI.Analysis.LiebThirring.Main
+import OAI.Analysis.LiebThirring.Multiplication
+import OAI.Analysis.LiebThirring.NegativeResolvent
+import OAI.Analysis.LiebThirring.OneState
+import OAI.Analysis.LiebThirring.OperatorForm
+import OAI.Analysis.LiebThirring.OperatorGraph
+import OAI.Analysis.LiebThirring.PotentialCompactness
+import OAI.Analysis.LiebThirring.PotentialSplit
+import OAI.Analysis.LiebThirring.Resolvent
+import OAI.Analysis.LiebThirring.SchrodingerOperator
+import OAI.Analysis.LiebThirring.SobolevBound
+import OAI.Analysis.LiebThirring.SpectralMain
+import OAI.Analysis.LiebThirring.SpectralMoment
+import OAI.Analysis.LiebThirring.WeightedCutoff
+import OAI.Analysis.LienardCycles.Main
+import OAI.Analysis.Lipschitz.TriangularLattice
+import OAI.Analysis.LipschitzEquivalence.Main
+import OAI.Analysis.LipschitzFree.Main
+import OAI.Analysis.LipschitzFree.Renorming
+import OAI.Analysis.LipschitzHilbert.Main
+import OAI.Analysis.Littlewood.Main
+import OAI.Analysis.Littlewood.FiniteFlatness
+import OAI.Analysis.LpDimension.Main
+import OAI.Analysis.Mahler
+import OAI.Analysis.Mahler.Classification
+import OAI.Analysis.MarkovType.All
+import OAI.Analysis.MassAction.Main
+import OAI.Analysis.MatrixSigning.Main
+import OAI.Analysis.MetricEntropy.Main
+import OAI.Analysis.ModularRecovery.Main
+import OAI.Analysis.MumfordShah.CompactExclusion
+import OAI.Analysis.MutuallyUnbiased.Bases.ThreeMain
+import OAI.Analysis.MutuallyUnbiased.Main
+import OAI.Analysis.Naimark.Tracial
+import OAI.Analysis.NavierStokes.VelocitySpace
+import OAI.Analysis.NefAdjoint.FractionalDensity
+import OAI.Analysis.NefAdjoint.SectionGauge
+import OAI.Analysis.NodalLength.Main
+import OAI.Analysis.Nonexpansive
+import OAI.Analysis.NuclearUltrapower.Main
+import OAI.Analysis.NumericalRange.Main
+import OAI.Analysis.PathSelection.Main
+import OAI.Analysis.PeriodicLattice.Main
+import OAI.Analysis.PlanarEndpoint.Main
+import OAI.Analysis.Polynomial.Exhaustion
+import OAI.Analysis.ProductSpaces.Main
+import OAI.Analysis.Pureness.Main
+import OAI.Analysis.Quantum.DimensionTen.Main
+import OAI.Analysis.RCDHeat.Main
+import OAI.Analysis.RecursivePotentials.EquivalentNorms
+import OAI.Analysis.RecursivePotentials.JoinedGeometry
+import OAI.Analysis.RecursivePotentials.Main
+import OAI.Analysis.RecursivePotentials.Model
+import OAI.Analysis.RecursivePotentials.TreeGeometry
+import OAI.Analysis.RecursivePotentials.VariableCompletion
+import OAI.Analysis.RecursivePotentials.VariableEstimates
+import OAI.Analysis.RecursivePotentials.VariableLp
+import OAI.Analysis.RecursivePotentials.VariableStability
+import OAI.Analysis.RecursivePotentials.ZeroRootGeometry
+import OAI.Analysis.RelativeGeneration.Main
+import OAI.Analysis.RieszRectifiability.Main
+import OAI.Analysis.RotationWeights.Main
+import OAI.Analysis.SegmentLenses.HeadRanges
+import OAI.Analysis.SegmentLenses.Main
+import OAI.Analysis.SeparableQuotients.Main
+import OAI.Analysis.SeparableQuotients.Positive.Main
+import OAI.Analysis.SignedDisk.Main
+import OAI.Analysis.SphereIsometry.Main
+import OAI.Analysis.Steinitz.Main
+import OAI.Analysis.StrictMeans.Main
+import OAI.Analysis.StructuralCrouzeix.Endpoint
+import OAI.Analysis.SymmetricDomains.Main
+import OAI.Analysis.TemperedDistribution.ConformalIdentities
+import OAI.Analysis.TraceCone.Main
+import OAI.Analysis.TracialCohomology.Vanishing
+import OAI.Analysis.TracialSplitting.Main
+import OAI.Analysis.TreePotential.Main
+import OAI.Analysis.Triangular.Main
+import OAI.Analysis.TriangularHilbert.Main
+import OAI.Analysis.Unitarizability.AllDiscrete
+import OAI.Analysis.Unitarizability.UniverseTransport
+import OAI.Analysis.VlasovMaxwell.Main
+import OAI.Analysis.WeakInfiniteness.Main
+import OAI.CategoryTheory.Globular.Main
+import OAI.CategoryTheory.ThickClosure.Main
+import OAI.CategoryTheory.Thomason.RankedFaithfulness
+import OAI.Combinatorics.Automata.Main
+import OAI.Combinatorics.BalancedRyser.Main
+import OAI.Combinatorics.BooleanFunctions.Main
+import OAI.Combinatorics.Chromatic.Main
+import OAI.Combinatorics.CliqueFree.Main
+import OAI.Combinatorics.Coboundary.Main
+import OAI.Combinatorics.CompleteCrossing.Main
+import OAI.Combinatorics.ContingencyTables.FactorialLift
+import OAI.Combinatorics.Crossing.Main
+import OAI.Combinatorics.CycleDecomposition.Main
+import OAI.Combinatorics.DiscreteConvexity.Main
+import OAI.Combinatorics.Distances.PolynomialIsolation
+import OAI.Combinatorics.EditDistance.Main
+import OAI.Combinatorics.EuclideanRamsey.Circle
+import OAI.Combinatorics.EuclideanRamsey.Main
+import OAI.Combinatorics.EuclideanRamsey.NineStatement
+import OAI.Combinatorics.EuclideanRamsey.Quadratic
+import OAI.Combinatorics.EuclideanRamsey.Spherical
+import OAI.Combinatorics.EuclideanRamsey.Transitive
+import OAI.Combinatorics.ExpectationThreshold.Main
+import OAI.Combinatorics.FiniteCircle.Main
+import OAI.Combinatorics.GotsmanLinial.Main
+import OAI.Combinatorics.GraphSpectrum.SampledChromatic
+import OAI.Combinatorics.GraphThreshold.Main
+import OAI.Combinatorics.HadwigerCounterexample.Main
+import OAI.Combinatorics.HalvingLines.Main
+import OAI.Combinatorics.Hamiltonian.Main
+import OAI.Combinatorics.Honeycomb.FiniteCylinder
+import OAI.Combinatorics.Honeycomb.PolygonMoment
+import OAI.Combinatorics.HoneycombBridges.ExactLength
+import OAI.Combinatorics.HoneycombBridges.Geometry
+import OAI.Combinatorics.HoneycombChords
+import OAI.Combinatorics.HoneycombCorridor.CorridorTransfer
+import OAI.Combinatorics.IndependentSets.Main
+import OAI.Combinatorics.InfiniteMatroid.Corollaries
+import OAI.Combinatorics.InfiniteMatroid.Main
+import OAI.Combinatorics.KMedian.Main
+import OAI.Combinatorics.KMedian.Threshold
+import OAI.Combinatorics.KMedianRecovery.Main
+import OAI.Combinatorics.KMedianRecovery.Refined.Main
+import OAI.Combinatorics.KMedianRecovery.Refined.Recovery
+import OAI.Combinatorics.KServer.Main
+import OAI.Combinatorics.ListHadwiger.Main
+import OAI.Combinatorics.MatchingCount.Main
+import OAI.Combinatorics.MatchingEntropy.Main
+import OAI.Combinatorics.Permanent.Main
+import OAI.Combinatorics.PolynomialVacuum.Main
+import OAI.Combinatorics.PortMatching.Reduction
+import OAI.Combinatorics.Progressions.Main
+import OAI.Combinatorics.Ramsey.CycleClique.Main
+import OAI.Combinatorics.Ramsey.Hypercube
+import OAI.Combinatorics.RamseyFive.Main
+import OAI.Combinatorics.StableGraphs.ExteriorBounds
+import OAI.Combinatorics.StripMass.Main
+import OAI.Combinatorics.SumProduct.Alignment.Main
+import OAI.Combinatorics.ThinTrees.AlgorithmicMain
+import OAI.Combinatorics.TriangleFace.Main
+import OAI.Combinatorics.MatchingPSD.AffineLift
+import OAI.Combinatorics.MatchingPSD.Main
+import OAI.Combinatorics.MatrixRemoval.Main
+import OAI.Combinatorics.MatroidCounting.CommonBases
+import OAI.Combinatorics.ParityLifts.AsymptoticBounds
+import OAI.Combinatorics.PerfectMatching.AlgorithmPartition
+import OAI.Combinatorics.PerfectMatching.Ascent
+import OAI.Combinatorics.PerfectMatching.CompletePairs
+import OAI.Combinatorics.PerfectMatching.Completion
+import OAI.Combinatorics.PerfectMatching.CountCorrect
+import OAI.Combinatorics.PerfectMatching.DyadicEntropy
+import OAI.Combinatorics.PerfectMatching.DyadicEnvelope
+import OAI.Combinatorics.PerfectMatching.ElementaryCount
+import OAI.Combinatorics.PerfectMatching.EntropyUpper
+import OAI.Combinatorics.PerfectMatching.FeasibilityRepair
+import OAI.Combinatorics.PerfectMatching.ForcedReduction
+import OAI.Combinatorics.PerfectMatching.Main
+import OAI.Combinatorics.PerfectMatching.Model
+import OAI.Combinatorics.PerfectMatching.ObjectivePenalty
+import OAI.Combinatorics.PerfectMatching.ParameterBounds
+import OAI.Combinatorics.PerfectMatching.Partition
+import OAI.Combinatorics.PerfectMatching.Separable
+import OAI.Combinatorics.PerfectMatching.SeparableCount
+import OAI.Combinatorics.PerfectMatching.SeparableCovariance
+import OAI.Combinatorics.PerfectMatching.SeparableLower
+import OAI.Combinatorics.PerfectMatching.SizeBounds
+import OAI.Combinatorics.PerfectMatching.Violation
+import OAI.Combinatorics.PerfectMatching.WeightedSpectrum
+import OAI.Combinatorics.PlanarL1.Embedding
+import OAI.Combinatorics.ProgressionColoring.Main
+import OAI.Combinatorics.Ryser.Main
+import OAI.Combinatorics.Ryser.OddExtensions
+import OAI.Combinatorics.SecondNeighborhood.Main
+import OAI.Combinatorics.Sensitivity.Main
+import OAI.Combinatorics.SharpRamsey.Main
+import OAI.Combinatorics.SharpThreshold.Main
+import OAI.Combinatorics.Sidorenko.RandomHost
+import OAI.Combinatorics.SnakyCertificate.Main
+import OAI.Combinatorics.SparsestCut.Main
+import OAI.Combinatorics.SphericalRamsey
+import OAI.Combinatorics.SphericalRamsey.Result
+import OAI.Combinatorics.SquareDifference.Main
+import OAI.Combinatorics.ThinTrees.Main
+import OAI.Combinatorics.TreeEdit.Main
+import OAI.Combinatorics.TreewidthL1.Main
+import OAI.Combinatorics.TriangleRemoval.Main
+import OAI.Combinatorics.TwoWayAutomata.ExplicitFamily
+import OAI.Combinatorics.TwoWayAutomata.Main
+import OAI.Combinatorics.TwoWayAutomata.PolynomialBound
+import OAI.Combinatorics.UniformKServer.Main
+import OAI.Combinatorics.VariableWL
+import OAI.Combinatorics.YoungDiagram.RealCapacity
+import OAI.Combinatorics.YoungDiagram.RealLargeCapacity
+import OAI.Computability.BinPacking.Main
+import OAI.Computability.DegreeRigidity.Main
+import OAI.Computability.DepthThree
+import OAI.Computability.DeterministicSum.Main
+import OAI.Computability.DirectedFeedback.Main
+import OAI.Computability.FourierCircuit.Main
+import OAI.Computability.FourierTransform.Main
+import OAI.Computability.GeneralizedCircuits.Mesh
+import OAI.Computability.Logspace.Deterministic
+import OAI.Computability.Logspace.Equality
+import OAI.Computability.LoopMatching.Endpoint
+import OAI.Computability.MatchingCount.ArgRoutine
+import OAI.Computability.MatchingCount.BinaryAlgorithm
+import OAI.Computability.MatchingCount.BinaryParser
+import OAI.Computability.MatchingCount.BinarySolve
+import OAI.Computability.MatchingCount.CappedPower
+import OAI.Computability.MatchingCount.CutAlgebra
+import OAI.Computability.MatchingCount.DenseOracle
+import OAI.Computability.MatchingCount.DivisionRoutine
+import OAI.Computability.MatchingCount.FiniteStack
+import OAI.Computability.MatchingCount.FoldRoutine
+import OAI.Computability.MatchingCount.GuardRoutine
+import OAI.Computability.MatchingCount.IntegerRoutine
+import OAI.Computability.MatchingCount.IterateRoutine
+import OAI.Computability.MatchingCount.LengthRoutine
+import OAI.Computability.MatchingCount.ListNumeric
+import OAI.Computability.MatchingCount.ListRoutine
+import OAI.Computability.MatchingCount.MapRoutine
+import OAI.Computability.MatchingCount.OddCutParity
+import OAI.Computability.MatchingCount.OddCutUncross
+import OAI.Computability.MatchingCount.OddMincut
+import OAI.Computability.MatchingCount.OptionCases
+import OAI.Computability.MatchingCount.OracleBounds
+import OAI.Computability.MatchingCount.OrbitRoutine
+import OAI.Computability.MatchingCount.PairRoutine
+import OAI.Computability.MatchingCount.ParamRoutine
+import OAI.Computability.MatchingCount.ParserSupport
+import OAI.Computability.MatchingCount.PolyCoding
+import OAI.Computability.MatchingCount.ProjectionRoutine
+import OAI.Computability.MatchingCount.RawBitsBridge
+import OAI.Computability.MatchingCount.RawContraction
+import OAI.Computability.MatchingCount.RawContractionCompiler
+import OAI.Computability.MatchingCount.RawData
+import OAI.Computability.MatchingCount.RawFlow
+import OAI.Computability.MatchingCount.RawFlowCompiler
+import OAI.Computability.MatchingCount.RawFlowSize
+import OAI.Computability.MatchingCount.RawInput
+import OAI.Computability.MatchingCount.RawOddBounds
+import OAI.Computability.MatchingCount.RawOddCompiler
+import OAI.Computability.MatchingCount.RawOddCut
+import OAI.Computability.MatchingCount.RawOptimizer
+import OAI.Computability.MatchingCount.RawOptimizerCompiler
+import OAI.Computability.MatchingCount.RawOptimizerData
+import OAI.Computability.MatchingCount.RawOptimizerDataCompiler
+import OAI.Computability.MatchingCount.RawOptimizerRows
+import OAI.Computability.MatchingCount.RawOptimizerRowsCompiler
+import OAI.Computability.MatchingCount.RawPrimitives
+import OAI.Computability.MatchingCount.RawRoutine
+import OAI.Computability.MatchingCount.RawSearch
+import OAI.Computability.MatchingCount.RawSearchCompiler
+import OAI.Computability.MatchingCount.RawSearchSize
+import OAI.Computability.MatchingCount.RawSize
+import OAI.Computability.MatchingCount.RecursiveBounds
+import OAI.Computability.MatchingCount.RecursiveCoding
+import OAI.Computability.MatchingCount.RecursiveCompiler
+import OAI.Computability.MatchingCount.RecursiveEval
+import OAI.Computability.MatchingCount.ResidualSearch
+import OAI.Computability.MatchingCount.ScaledAlgorithm
+import OAI.Computability.MatchingCount.ScaledBounds
+import OAI.Computability.MatchingCount.ScaledState
+import OAI.Computability.MatchingCount.SelectRoutine
+import OAI.Computability.MatchingCount.SequenceRoutine
+import OAI.Computability.MatchingCount.StackCopy
+import OAI.Computability.MatchingCount.StackEmbed
+import OAI.Computability.MatchingCount.StackFrame
+import OAI.Computability.MatchingCount.StackMacros
+import OAI.Computability.MatchingCount.StackPair
+import OAI.Computability.MatchingCount.StackRoutine
+import OAI.Computability.MatchingCount.SumRoutine
+import OAI.Computability.MatchingCount.TerminalMincut
+import OAI.Computability.MatchingCount.UnaryArithmetic
+import OAI.Computability.MatchingCount.UnaryClog
+import OAI.Computability.MatchingCount.UnaryDivision
+import OAI.Computability.MatchingCount.UnitFlow
+import OAI.Computability.MatchingCount.UnitMincut
+import OAI.Computability.MaxCut.Main
+import OAI.Computability.MeanPayoff.Main
+import OAI.Computability.MinUncut.Main
+import OAI.Computability.PerfectCompleteness.Main
+import OAI.Computability.QuantumFactoring.Main
+import OAI.Computability.RandomMean.Main
+import OAI.Computability.RationalHitting.Main
+import OAI.Computability.Scheduling.Main
+import OAI.Computability.SpaceSimulation.Main
+import OAI.Computability.StarHeight.Main
+import OAI.Computability.Superstring.Main
+import OAI.Computability.TypeSystem.Normalization
+import OAI.Computability.UniqueGames.Main
+import OAI.Computability.VertexCover.Main
+import OAI.Computability.WLIdentification.Main
+import OAI.Computability.WeisfeilerLeman.Main
+import OAI.Computability.WitnessedChoice.Main
+import OAI.Dynamics.ConditionalShuffle.TwoColors
+import OAI.Dynamics.EntropyCounterexample
+import OAI.Dynamics.MultipleMixing.Main
+import OAI.Dynamics.SmoothObstruction.Main
+import OAI.Dynamics.StandardMap.ComponentsMain
+import OAI.Dynamics.StandardMap.EntropyEndpoint
+import OAI.Dynamics.StandardMap.EntropyLyapunov
+import OAI.Dynamics.ThreeTorus.Main
+import OAI.Dynamics.TriangleBilliards.Main
+import OAI.GameTheory.SnakyConditional.Main
+import OAI.GameTheory.SnakyTwentyOne.Main
+import OAI.Geometry.Anticanonical.BranchedQuotient
+import OAI.Geometry.Anticanonical.DeckRecovery
+import OAI.Geometry.Anticanonical.DeletedFaces
+import OAI.Geometry.Anticanonical.HilbertEstimates
+import OAI.Geometry.Arnold.Main
+import OAI.Geometry.BallPacking.Main
+import OAI.Geometry.BallPacking.Necessity.Main
+import OAI.Geometry.Borsuk.Main
+import OAI.Geometry.CAT0Fillings.Main
+import OAI.Geometry.Cannon.Main
+import OAI.Geometry.ConjugatePoints.ThreeManifold
+import OAI.Geometry.CoveringDensity.Main
+import OAI.Geometry.CubicTorus.Main
+import OAI.Geometry.CylinderCovering.Main
+import OAI.Geometry.DoublingHilbert.CompactBanach
+import OAI.Geometry.DoublingHilbert.Main
+import OAI.Geometry.HarmonicGrowth.Main
+import OAI.Geometry.HeilbronnTriangle
+import OAI.Geometry.HyperbolicGroups.BipartiteLink
+import OAI.Geometry.HyperbolicGroups.ClosedPicture
+import OAI.Geometry.HyperbolicGroups.LetterFillingArea
+import OAI.Geometry.HyperbolicGroups.LocalSegmentContinuation
+import OAI.Geometry.HyperbolicGroups.LocalSegmentTools
+import OAI.Geometry.HyperbolicGroups.Main
+import OAI.Geometry.HyperbolicGroups.Parity
+import OAI.Geometry.HyperbolicGroups.PolygonContraction
+import OAI.Geometry.HyperbolicGroups.TreePathDepth
+import OAI.Geometry.HypersurfaceGerms.Main
+import OAI.Geometry.Immersion.ClosedSurface.Main
+import OAI.Geometry.IntegralFillings.Main
+import OAI.Geometry.IsometricImmersion.All
+import OAI.Geometry.Kahler
+import OAI.Geometry.KahlerSplitting.Main
+import OAI.Geometry.LatticeCovering
+import OAI.Geometry.LogVolume.BrunnMinkowski
+import OAI.Geometry.Manifold.ProjectiveSphere
+import OAI.Geometry.NearbyLagrangian.Main
+import OAI.Geometry.NearbyLagrangian.TubeKernel
+import OAI.Geometry.NegativeCurvature.Tube
+import OAI.Geometry.NilpotentCharts.Main
+import OAI.Geometry.NodalSets.Main
+import OAI.Geometry.Perimeter.Regularity.Main
+import OAI.Geometry.PeriodicTiling.Main
+import OAI.Geometry.PinnedDistances.Pins
+import OAI.Geometry.PlaneColoring.Five
+import OAI.Geometry.PlaneColoring.Main
+import OAI.Geometry.PolarProducts.Main
+import OAI.Geometry.PrescribedPotential.AtlasLocalization
+import OAI.Geometry.PrescribedPotential.AugmentedVolume
+import OAI.Geometry.PrescribedPotential.CalabiFrameCalculus
+import OAI.Geometry.PrescribedPotential.CalabiFrameScalars
+import OAI.Geometry.PrescribedPotential.CalabiRicciContraction
+import OAI.Geometry.PrescribedPotential.ClosedRangeSmooth
+import OAI.Geometry.PrescribedPotential.CommutatorSupport
+import OAI.Geometry.PrescribedPotential.CompletedProduct
+import OAI.Geometry.PrescribedPotential.CompletedRealification
+import OAI.Geometry.PrescribedPotential.CompletedVolume
+import OAI.Geometry.PrescribedPotential.CoordinateEllipticity
+import OAI.Geometry.PrescribedPotential.CoordinateExtension
+import OAI.Geometry.PrescribedPotential.CoordinateMinimumBound
+import OAI.Geometry.PrescribedPotential.CoordinatePotential
+import OAI.Geometry.PrescribedPotential.DerivativeEllipticGain
+import OAI.Geometry.PrescribedPotential.DerivativeLocalization
+import OAI.Geometry.PrescribedPotential.EvenRellich
+import OAI.Geometry.PrescribedPotential.FixedScaleBootstrap
+import OAI.Geometry.PrescribedPotential.FredholmKernel
+import OAI.Geometry.PrescribedPotential.FredholmRegularity
+import OAI.Geometry.PrescribedPotential.GlobalBounds
+import OAI.Geometry.PrescribedPotential.GlobalError
+import OAI.Geometry.PrescribedPotential.GlobalFiniteEmbedding
+import OAI.Geometry.PrescribedPotential.GlobalNonlinearCommutator
+import OAI.Geometry.PrescribedPotential.GlobalProduct
+import OAI.Geometry.PrescribedPotential.GlobalRellich
+import OAI.Geometry.PrescribedPotential.GlobalSmooth
+import OAI.Geometry.PrescribedPotential.HigherMetricOperator
+import OAI.Geometry.PrescribedPotential.HigherPoisson
+import OAI.Geometry.PrescribedPotential.HigherResolvent
+import OAI.Geometry.PrescribedPotential.LinearizedCommutator
+import OAI.Geometry.PrescribedPotential.MetricQuasilinearEquation
+import OAI.Geometry.PrescribedPotential.NonlinearCoreCommutator
+import OAI.Geometry.PrescribedPotential.NonlinearRemainderStrong
+import OAI.Geometry.PrescribedPotential.PathCalabiBound
+import OAI.Geometry.PrescribedPotential.PathCoordinateDensity
+import OAI.Geometry.PrescribedPotential.PathFrameBounds
+import OAI.Geometry.PrescribedPotential.PointPoisson
+import OAI.Geometry.PrescribedPotential.PositiveOpenness
+import OAI.Geometry.PrescribedPotential.PositivePotentialLinearization
+import OAI.Geometry.PrescribedPotential.PotentialDensityDifferential
+import OAI.Geometry.PrescribedPotential.PrescribedPotential
+import OAI.Geometry.PrescribedPotential.QuantitativeResolvent
+import OAI.Geometry.PrescribedPotential.RealMapSmooth
+import OAI.Geometry.PrescribedPotential.RealPoisson
+import OAI.Geometry.PrescribedPotential.RealSmoothCompactness
+import OAI.Geometry.PrescribedPotential.RegularityCutoffs
+import OAI.Geometry.PrescribedPotential.RegularityJets
+import OAI.Geometry.PrescribedPotential.SmoothChartInverse
+import OAI.Geometry.PrescribedPotential.SmoothLocalPath
+import OAI.Geometry.PrescribedPotential.SmoothPathLimit
+import OAI.Geometry.PrescribedPotential.SmoothSequenceCompactness
+import OAI.Geometry.PrescribedPotential.SobolevDetDifferential
+import OAI.Geometry.PrescribedPotential.SobolevExponential
+import OAI.Geometry.PrescribedPotential.SobolevFirstGain
+import OAI.Geometry.PrescribedPotential.SobolevOpenness
+import OAI.Geometry.PrescribedPotential.TwoScaleRegularity
+import OAI.Geometry.PrescribedPotential.UniformPathSobolev
+import OAI.Geometry.PrescribedPotential.WeakDerivativeLocalization
+import OAI.Geometry.PrescribedPotential.WeakPositive
+import OAI.Geometry.PrescribedRicci.BoundedSmoothPoisson
+import OAI.Geometry.PrescribedRicci.CalabiGradient
+import OAI.Geometry.PrescribedRicci.CalabiMetricC1
+import OAI.Geometry.PrescribedRicci.CalabiRough
+import OAI.Geometry.PrescribedRicci.CalabiTrace
+import OAI.Geometry.PrescribedRicci.CalabiTraceBound
+import OAI.Geometry.PrescribedRicci.ChernBianchi
+import OAI.Geometry.PrescribedRicci.ChernLuConnection
+import OAI.Geometry.PrescribedRicci.ChernRough
+import OAI.Geometry.PrescribedRicci.CofactorCore
+import OAI.Geometry.PrescribedRicci.CofactorCoreEquation
+import OAI.Geometry.PrescribedRicci.CofactorLocalization
+import OAI.Geometry.PrescribedRicci.CompactPathBounds
+import OAI.Geometry.PrescribedRicci.CompactPathRicci
+import OAI.Geometry.PrescribedRicci.FixedCofactorBootstrap
+import OAI.Geometry.PrescribedRicci.FixedCofactorStep
+import OAI.Geometry.PrescribedRicci.GlobalSmoothCompact
+import OAI.Geometry.PrescribedRicci.IntegratedKahlerEnergy
+import OAI.Geometry.PrescribedRicci.JetProductL2
+import OAI.Geometry.PrescribedRicci.KahlerCauchySchwarz
+import OAI.Geometry.PrescribedRicci.KahlerChartGradient
+import OAI.Geometry.PrescribedRicci.KahlerChartLp
+import OAI.Geometry.PrescribedRicci.KahlerEnergyProduct
+import OAI.Geometry.PrescribedRicci.KahlerIntegralPos
+import OAI.Geometry.PrescribedRicci.KahlerL2Comparison
+import OAI.Geometry.PrescribedRicci.KahlerPoincare
+import OAI.Geometry.PrescribedRicci.KahlerPoissonL2
+import OAI.Geometry.PrescribedRicci.KahlerSobolev
+import OAI.Geometry.PrescribedRicci.KahlerVolumeMeasure
+import OAI.Geometry.PrescribedRicci.KahlerVolumeVariation
+import OAI.Geometry.PrescribedRicci.MongeAmpereC0
+import OAI.Geometry.PrescribedRicci.MongeAmpereEnergy
+import OAI.Geometry.PrescribedRicci.MongeAmpereL2
+import OAI.Geometry.PrescribedRicci.NormalizeVolumePath
+import OAI.Geometry.PrescribedRicci.PathLocalGain
+import OAI.Geometry.PrescribedRicci.PathUniformHigher
+import OAI.Geometry.PrescribedRicci.PathUniformSobolevBase
+import OAI.Geometry.PrescribedRicci.PositiveOpenness
+import OAI.Geometry.PrescribedRicci.PositivePathLimit
+import OAI.Geometry.PrescribedRicci.PrescribedRicci
+import OAI.Geometry.PrescribedRicci.SmoothLocalInverse
+import OAI.Geometry.PrescribedRicci.SmoothPathLimit
+import OAI.Geometry.PrescribedRicci.SmoothPositivePath
+import OAI.Geometry.PrescribedRicci.TameCofactorGain
+import OAI.Geometry.PrescribedRicci.TraceMaximum
+import OAI.Geometry.PrescribedRicci.UniformCalabi
+import OAI.Geometry.PrescribedRicci.UniformChernLu
+import OAI.Geometry.PrescribedRicci.UniformMetricBounds
+import OAI.Geometry.PrescribedRicci.UniformPathPatches
+import OAI.Geometry.PrescribedRicci.WeightedMongeAmpereEnergy
+import OAI.Geometry.ProjectionBodies.Main
+import OAI.Geometry.ProjectionBody.Main
+import OAI.Geometry.ProjectionVolume.Main
+import OAI.Geometry.QuadricBundles.Main
+import OAI.Geometry.Riemannian.HarmonicCore.Main
+import OAI.Geometry.RuledCovering.Main
+import OAI.Geometry.SmoothYau.Main
+import OAI.Geometry.SplitTangent.AllCompactLeaves
+import OAI.Geometry.SplitTangent.AllProjectiveLeaves
+import OAI.Geometry.SplitTangent.AnalyticFiniteRelations
+import OAI.Geometry.SplitTangent.ArbitraryPlaqueLeaves
+import OAI.Geometry.SplitTangent.BasicDerivativeKernel
+import OAI.Geometry.SplitTangent.BoxFirstTransport
+import OAI.Geometry.SplitTangent.ChartSubmersion
+import OAI.Geometry.SplitTangent.CompactLeafQuotient
+import OAI.Geometry.SplitTangent.CompactPositiveLeaf
+import OAI.Geometry.SplitTangent.ComplementaryLeafIntersections
+import OAI.Geometry.SplitTangent.DenseCompactLeaves
+import OAI.Geometry.SplitTangent.DenseRationalVariation
+import OAI.Geometry.SplitTangent.EmbeddedLeafCharts
+import OAI.Geometry.SplitTangent.EmbeddedRationalCurve
+import OAI.Geometry.SplitTangent.EmbeddedTangentLift
+import OAI.Geometry.SplitTangent.EverywhereSmoothLeafQuotient
+import OAI.Geometry.SplitTangent.GenericLeafProductCover
+import OAI.Geometry.SplitTangent.GlobalAffineRelations
+import OAI.Geometry.SplitTangent.GlobalLeafRelations
+import OAI.Geometry.SplitTangent.GlobalProducts
+import OAI.Geometry.SplitTangent.GlobalTopologicalProduct
+import OAI.Geometry.SplitTangent.HolomorphicBasicTransport
+import OAI.Geometry.SplitTangent.HolomorphicLeafBasic
+import OAI.Geometry.SplitTangent.HolomorphicManifoldProduct
+import OAI.Geometry.SplitTangent.IntegrabilityReduction
+import OAI.Geometry.SplitTangent.IntrinsicPositiveLeaf
+import OAI.Geometry.SplitTangent.LeafAffineRelations
+import OAI.Geometry.SplitTangent.LeafAlgebraicDependence
+import OAI.Geometry.SplitTangent.LeafClosureRelations
+import OAI.Geometry.SplitTangent.LeafCompactLimits
+import OAI.Geometry.SplitTangent.LeafGermIdentity
+import OAI.Geometry.SplitTangent.LeafHomogeneousRelations
+import OAI.Geometry.SplitTangent.LeafLimitRelations
+import OAI.Geometry.SplitTangent.LeafLimitsSinglePlaque
+import OAI.Geometry.SplitTangent.LeafPolynomialGrowth
+import OAI.Geometry.SplitTangent.LeafProducts
+import OAI.Geometry.SplitTangent.LeafTopology
+import OAI.Geometry.SplitTangent.ManifoldJetAnalytic
+import OAI.Geometry.SplitTangent.OpenPositiveLeaves
+import OAI.Geometry.SplitTangent.PlaqueConnectedSets
+import OAI.Geometry.SplitTangent.PlaqueTangentImage
+import OAI.Geometry.SplitTangent.PolynomialLeafGerms
+import OAI.Geometry.SplitTangent.PositiveCoverLift
+import OAI.Geometry.SplitTangent.ProductCriteria
+import OAI.Geometry.SplitTangent.ProjectiveHypersurfaces
+import OAI.Geometry.SplitTangent.ProjectivePositiveLeaf
+import OAI.Geometry.SplitTangent.ProjectiveTangentBox
+import OAI.Geometry.SplitTangent.QuotientSubmersions
+import OAI.Geometry.SplitTangent.RCCovariantVanishing
+import OAI.Geometry.SplitTangent.RationalCoverLift
+import OAI.Geometry.SplitTangent.RationalReparam
+import OAI.Geometry.SplitTangent.RationalVariation
+import OAI.Geometry.SplitTangent.TangentCurveLeaf
+import OAI.Geometry.SplitTangent.TangentRationalChains
+import OAI.Geometry.SplitTangent.TransverseDiscrete
+import OAI.Geometry.SplitTangent.TransverseQuotient
+import OAI.Geometry.SplitTangent.ZariskiPairOpen
+import OAI.Geometry.Symplectic.MorseBounds.Main
+import OAI.Geometry.TamingCompatibility.Main
+import OAI.Geometry.TranslativeCovering.Main
+import OAI.Geometry.TriangularCovering.Main
+import OAI.Geometry.UnitDistances.Main
+import OAI.Geometry.Varifold.NullComparison
+import OAI.Geometry.WeakHessian.Main
+import OAI.Geometry.WeakMTW.Main
+import OAI.GroupTheory.ArithmeticGerms.Stabilizers
+import OAI.GroupTheory.ArtinCAT0.Main
+import OAI.GroupTheory.BooneHigman.Main
+import OAI.GroupTheory.CyclicSeries.Length
+import OAI.GroupTheory.FiniteIndex.Realization
+import OAI.GroupTheory.FiniteKernel.VirtualAbelian
+import OAI.GroupTheory.FiniteLattice.LabelledCollection
+import OAI.GroupTheory.FiniteLattice.NormalPreimage
+import OAI.GroupTheory.FiniteType.Main
+import OAI.GroupTheory.Heisenberg.BarDetector
+import OAI.GroupTheory.Hyperbolic.Main
+import OAI.GroupTheory.Kervaire.Main
+import OAI.GroupTheory.PChains.Cancellation
+import OAI.GroupTheory.PeriodicGroups.Main
+import OAI.GroupTheory.SimpleAmenable.Main
+import OAI.GroupTheory.SimpleOvergroups.Main
+import OAI.GroupTheory.Thompson.Main
+import OAI.GroupTheory.UniversalGroup.Main
+import OAI.InformationTheory.AmplitudeDamping.Main
+import OAI.InformationTheory.BooleanNoise.Main
+import OAI.InformationTheory.DimensionTen.StateKey
+import OAI.InformationTheory.Entanglement.Main
+import OAI.InformationTheory.PhotonNumber.Inequality
+import OAI.InformationTheory.QuantumCircuit.Main
+import OAI.InformationTheory.QuantumCircuit.Regularity
+import OAI.InformationTheory.SecretKey.DimensionTen
+import OAI.InformationTheory.SecretKey.Main
+import OAI.InformationTheory.SoftChannel.Main
+import OAI.LinearAlgebra.Barker.Main
+import OAI.LinearAlgebra.CirculantHadamard.Main
+import OAI.LinearAlgebra.CliffordAlgebra.KugaSatake
+import OAI.LinearAlgebra.Descent.Rational
+import OAI.LinearAlgebra.Hadamard.Main
+import OAI.LinearAlgebra.MatrixFields.Main
+import OAI.LinearAlgebra.MatrixMultiplication.Main
+import OAI.LinearAlgebra.TensorProduct.ScalarDescent
+import OAI.MathematicalPhysics.AlternatingFlow.Development
+import OAI.MathematicalPhysics.Anderson.Certificates
+import OAI.MathematicalPhysics.BFSS.Main
+import OAI.MathematicalPhysics.Coulomb.OuterRadius
+import OAI.MathematicalPhysics.CriticalMixing.Main
+import OAI.MathematicalPhysics.CriticalSK.LowerExponents
+import OAI.MathematicalPhysics.CriticalSK.Main
+import OAI.MathematicalPhysics.Elasticity.Uniqueness
+import OAI.MathematicalPhysics.HardSphere.GrandCanonical
+import OAI.MathematicalPhysics.Heisenberg.Main
+import OAI.MathematicalPhysics.NavierStokes.BalancedTransport.MainResult
+import OAI.MathematicalPhysics.NavierStokes.Material.Main
+import OAI.MathematicalPhysics.NavierStokes.ShearFlows.Main
+import OAI.MathematicalPhysics.NavierStokes.VelocityDetection.Main
+import OAI.MathematicalPhysics.PEPSFilters.GeometricOptimizer
+import OAI.MathematicalPhysics.PEPSMove.PhysicalMove
+import OAI.MathematicalPhysics.PEPSSubvolume.Main
+import OAI.MathematicalPhysics.PlanarAnderson.Spectrum
+import OAI.MathematicalPhysics.PrefixFlows.Main
+import OAI.MathematicalPhysics.RapidForcing.Main
+import OAI.MathematicalPhysics.RecorderFlows.Main
+import OAI.MathematicalPhysics.SheetFlows.SheetProgram
+import OAI.MathematicalPhysics.TensorNetwork.VectorColumn
+import OAI.MeasureTheory.DyadicAvoidance.Main
+import OAI.MeasureTheory.Falconer.Main
+import OAI.MeasureTheory.FalconerCompletion.Main
+import OAI.MeasureTheory.SelfSimilar.Main
+import OAI.MeasureTheory.SelfSimilar.DimensionCorollaries
+import OAI.ModelTheory.AbstractElementary.ModelExistence
+import OAI.ModelTheory.Categoricity.Main
+import OAI.ModelTheory.Choiceless.Separation
+import OAI.NumberTheory.Catalan.Main
+import OAI.NumberTheory.CubicGauss.Main
+import OAI.NumberTheory.CubicGram.Main
+import OAI.NumberTheory.DirichletL.Hecke.Nonvanishing
+import OAI.NumberTheory.DirichletL.Nonvanishing
+import OAI.NumberTheory.DuffinSchaeffer.Reductions
+import OAI.NumberTheory.EgyptianFractions.All
+import OAI.NumberTheory.GaussianMoat.Main
+import OAI.NumberTheory.Jacobsthal.Main
+import OAI.NumberTheory.JointDickman.PaperMain
+import OAI.NumberTheory.NumberField.UnitRadicals
+import OAI.NumberTheory.OrdinaryCorrelations.AbsoluteDefect.Main
+import OAI.NumberTheory.OrdinaryCorrelations.Elliott.Main
+import OAI.NumberTheory.OrdinaryCorrelations.HighTrace.Main
+import OAI.NumberTheory.Ostmann.Main
+import OAI.NumberTheory.PiExponent.Main
+import OAI.NumberTheory.PowerFree.Main
+import OAI.NumberTheory.PrimeGaps.RatioCorollary
+import OAI.NumberTheory.PrimePackets.Main
+import OAI.NumberTheory.QuadraticForms.ComponentLengths
+import OAI.NumberTheory.ShortEgyptian.Main
+import OAI.NumberTheory.SiegelZeros.Main
+import OAI.NumberTheory.SingleFold.Main
+import OAI.NumberTheory.TotientAsymptotic.Main
+import OAI.NumberTheory.TotientFibers.FiniteFiber
+import OAI.Optimization.Tropical.TruffetCounterexample
+import OAI.Probability.Ballisticity.Main
+import OAI.Probability.Ballisticity.VelocityHemisphere
+import OAI.Probability.BenjaminiSchramm
+import OAI.Probability.BinarySweep.Main
+import OAI.Probability.ClassicalON.Main
+import OAI.Probability.Contingency.LocalRepair
+import OAI.Probability.Contingency.PreparedResidual
+import OAI.Probability.ContinuumTransition.Main
+import OAI.Probability.CoordinateSweeps.Main
+import OAI.Probability.CriticalPercolation.Main
+import OAI.Probability.CriticalZ3.Main
+import OAI.Probability.CubeShuffle.WeightedMoments
+import OAI.Probability.DilutedSpin.Main
+import OAI.Probability.DirectionalWalk.Main
+import OAI.Probability.EntangledGames.Main
+import OAI.Probability.FKPressure.Main
+import OAI.Probability.FirstPassage.Main
+import OAI.Probability.FKMaps.Convergence
+import OAI.Probability.FreeIsing.Main
+import OAI.Probability.GaussianInformation.Main
+import OAI.Probability.GaussianPropeller.Main
+import OAI.Probability.GaussianRegression.Main
+import OAI.Probability.GaussianReplacement.Streaming
+import OAI.Probability.Honeycomb.FreeEnergy
+import OAI.Probability.Honeycomb.TriangleVisits
+import OAI.Probability.Ising.Main
+import OAI.Probability.IsingPerceptron.FullReplicaIBP
+import OAI.Probability.IsingPerceptron.VariationalFiniteness
+import OAI.Probability.JammingConcavity.DeletionUniformRowEstimates
+import OAI.Probability.JammingConcavity.RowFiniteDiagonalRowGeneralInterpolation
+import OAI.Probability.JammingConcavity.RowQuadraticTerminalLimit
+import OAI.Probability.JammingConcavity.VarianceTensorization
+import OAI.Probability.LatticeSurface.Brownian.BrownianClockFacet
+import OAI.Probability.LatticeSurface.Brownian.BrownianClockSupport
+import OAI.Probability.LatticeSurface.Brownian.BrownianPath
+import OAI.Probability.LatticeSurface.Brownian.BrownianPathIdentities
+import OAI.Probability.LatticeSurface.Coercivity.HoleWeighted
+import OAI.Probability.LatticeSurface.Convexity.ClusterDynamicsDefs
+import OAI.Probability.LatticeSurface.Convexity.FiniteLevelBarrier
+import OAI.Probability.LatticeSurface.Convexity.Metropolis
+import OAI.Probability.LatticeSurface.Convexity.MetropolisSupport
+import OAI.Probability.LatticeSurface.Convexity.ProductSums
+import OAI.Probability.LatticeSurface.Convexity.SupportUnit
+import OAI.Probability.LatticeSurface.Convexity.TailNumerics
+import OAI.Probability.LatticeSurface.Convexity.TransportResampling
+import OAI.Probability.LatticeSurface.Dynamics.CompactCoupling
+import OAI.Probability.LatticeSurface.Dynamics.LatticeShift
+import OAI.Probability.LatticeSurface.Dynamics.ProjectionComparison
+import OAI.Probability.LatticeSurface.Dynamics.ProjectionGlobal
+import OAI.Probability.LatticeSurface.Dynamics.ProjectionLimit
+import OAI.Probability.LatticeSurface.Dynamics.TiltApplication
+import OAI.Probability.LatticeSurface.Dynamics.TiltSupport
+import OAI.Probability.LatticeSurface.Dynamics.WeightedInitial
+import OAI.Probability.LatticeSurface.Geometry.ContactVolume
+import OAI.Probability.LatticeSurface.Moments.Averages
+import OAI.Probability.Localization.Main
+import OAI.Probability.LogConcave.Main
+import OAI.Probability.MatroidProphet
+import OAI.Probability.MatroidSecretary.Main
+import OAI.Probability.MemoryPrecision.Main
+import OAI.Probability.ParisiFinite.ParisiCanonicalTransport
+import OAI.Probability.ParisiFinite.ParisiEntropyDecomposition
+import OAI.Probability.ParisiFinite.RealShiftNormSqLe
+import OAI.Probability.ParisiFinite.SpinFlip
+import OAI.Probability.ParisiFinite.WeightedGradientField
+import OAI.Probability.Perceptron.Main
+import OAI.Probability.PosteriorReplicas.Main
+import OAI.Probability.ProbabilityMeasure.ArctanCenter
+import OAI.Probability.ProjectionMoments.Main
+import OAI.Probability.RadialTransition.DensityInterval
+import OAI.Probability.RadialTransition.Main
+import OAI.Probability.RandomSAT.Main
+import OAI.Probability.Renewal.Main
+import OAI.Probability.SATComputability.Main
+import OAI.Probability.SATVariance.Main
+import OAI.Probability.SKBarriers.Main
+import OAI.Probability.SKCutoff.Denominator
+import OAI.Probability.SKFluctuations.ComparisonMinimumRealOverlapConclusion
+import OAI.Probability.SKGap.Main
+import OAI.Probability.SKSupport.Main
+import OAI.Probability.SKValue.Main
+import OAI.Probability.SLE.MainReduction
+import OAI.Probability.SLE.MomentGauge
+import OAI.Probability.SLE.SausageCharge
+import OAI.Probability.SLEGauge.Main
+import OAI.Probability.SLELocalization.CircleMeans
+import OAI.Probability.SLELocalization.DriverStability
+import OAI.Probability.SLELocalization.FilledObstacle
+import OAI.Probability.SLELocalization.LoewnerCocycle
+import OAI.Probability.SLELocalization.MappedExit
+import OAI.Probability.SLELocalization.MeshSlab
+import OAI.Probability.SLELocalization.PastObstacle
+import OAI.Probability.SLELocalization.RetainedFirstMoment
+import OAI.Probability.SLELocalization.SlitMaps
+import OAI.Probability.SLELocalization.SpatialLimit
+import OAI.Probability.SLELocalization.StrongRestart
+import OAI.Probability.SLELocalization.TraceAvoidance
+import OAI.Probability.SLELocalization.TruncatedEnergy
+import OAI.Probability.SignedSweeps.Main
+import OAI.Probability.SpanningForest.Main
+import OAI.Probability.SphericalField.Main
+import OAI.Probability.SphericalPerceptron.PressureComparison
+import OAI.Probability.StrongRayleigh
+import OAI.Probability.Subsphere.Main
+import OAI.Probability.SwitchChain
+import OAI.Probability.ThorpCompatibility.Main
+import OAI.Probability.ThorpResults.FullDevelopment
+import OAI.Probability.ThorpRouting.Main
+import OAI.Probability.ThorpRouting.RegularTrace
+import OAI.Probability.ThorpShuffle.FirstFourier
+import OAI.Probability.ThorpShuffle.FirstMain
+import OAI.Probability.ThorpShuffle.FixedLists.Main
+import OAI.Probability.ThorpShuffle.Main
+import OAI.Probability.ThreeState.Main
+import OAI.Probability.ThreeStateClauses.Main
+import OAI.Probability.TraceReconstruction
+import OAI.Probability.UnimodularPercolation.CayleyMain
+import OAI.Probability.UnimodularPercolation.Main
+import OAI.Probability.UnimodularPercolation.UnimodularMain
+import OAI.Probability.Voronoi.CardyFormula
+import OAI.RepresentationTheory.AlperinWeights.Brauer
+import OAI.RepresentationTheory.Block.Defect
+import OAI.RepresentationTheory.FoulkesHowe
+import OAI.RepresentationTheory.FoulkesSixth.Main
+import OAI.RepresentationTheory.GraphModels.NormalActions
+import OAI.RepresentationTheory.KazhdanLusztig.Invariance
+import OAI.RepresentationTheory.PartialPermutation.Main
+import OAI.RepresentationTheory.ProjectiveInduction.CosetTransport
+import OAI.RepresentationTheory.RestrictedTilting.Main
+import OAI.RepresentationTheory.RowColumn.Occupied
+import OAI.RepresentationTheory.Saxl.Main
+import OAI.RepresentationTheory.SpinAngle.TypeAngle
+import OAI.RepresentationTheory.TensorSquares.Main
+import OAI.RepresentationTheory.Torus.ConvexWeights
+import OAI.RepresentationTheory.VertexAlgebra.Main
+import OAI.RingTheory.BassTrace.Main
+import OAI.RingTheory.BassTrace.TorsionFree
+import OAI.RingTheory.DirectFiniteness.FinitelyPresented
+import OAI.RingTheory.DirectFiniteness.Main
+import OAI.RingTheory.DivisionRing.FiniteDescent
+import OAI.RingTheory.FormalGroup.Main
+import OAI.RingTheory.GroupAction.TraceOne
+import OAI.RingTheory.Kurosh.Main
+import OAI.RingTheory.Multiplicity.Main
+import OAI.RingTheory.Tachikawa.Counterexample
+import OAI.SetTheory.PartitionConsistency.Main
+import OAI.SetTheory.PartitionPrinciple.Main
+import OAI.Topology.ArtinGroups.Main
+import OAI.Topology.ArtinGroups.ParabolicIntersections.Main
+import OAI.Topology.EilenbergGanea.Main
+import OAI.Topology.TorusCompression.Boundary
+import OAI.Topology.CoarseAssembly
+import OAI.RepresentationTheory.UniversalSquare.Main
+import OAI.Probability.SKRatio.Main
+import OAI.Geometry.Relativity.CKS.Main
+import OAI.Geometry.SurfaceImmersion.All
+
+import OAI.GroupTheory.PolycyclicRecognition

lean/OAI/Topology/CoarseAssembly.lean

@@ -1,190 +1,190 @@
-import OAI.Topology.CoarseAssembly.FiberedPersistence
-import OAI.Topology.CoarseAssembly.HomotopyFilling
-import OAI.Topology.CoarseAssembly.LatticeHat
-import OAI.Topology.CoarseAssembly.FilledDisc
-import OAI.Topology.CoarseAssembly.FilledEdges
-import OAI.Topology.CoarseAssembly.FilledWalks
-import OAI.Topology.CoarseAssembly.CompactFamilyMap
-import OAI.Topology.CoarseAssembly.FamilyFilling
-import OAI.Topology.CoarseAssembly.RipsProperEmbedding
-import OAI.Topology.CoarseAssembly.RoeInclusion
-import OAI.Topology.CoarseAssembly.BoundaryGeometry
-import OAI.Topology.CoarseAssembly.BoundaryWalks
-import OAI.Topology.CoarseAssembly.BoundaryComponents
-import OAI.Topology.CoarseAssembly.HexagonSides
-import OAI.Topology.CoarseAssembly.RipsMapCalculus
-import OAI.Topology.CoarseAssembly.RipsReindex
-import OAI.Topology.CoarseAssembly.ConcretePersistence
-import OAI.Topology.CoarseAssembly.BoundaryLoopMaps
-import OAI.Topology.CoarseAssembly.BoundaryAssemblyMaps
-import OAI.Topology.CoarseAssembly.BoundaryConclusion
-import OAI.Topology.CoarseAssembly.PublishedInputReduction
-import OAI.Topology.CoarseAssembly.CycleCompactPerturbation
-import OAI.Topology.CoarseAssembly.UnitalCycleNormalization
-import OAI.Topology.CoarseAssembly.FiniteCycle
-import OAI.Topology.CoarseAssembly.FredholmSquare
-import OAI.Topology.CoarseAssembly.UnitalCycleInvertible
-import OAI.Topology.CoarseAssembly.CompactFunctionalCalculus
-import OAI.Topology.CoarseAssembly.RipsEmbedding
-import OAI.Topology.CoarseAssembly.ClopenRestriction
-import OAI.Topology.CoarseAssembly.ClopenNaturality
-import OAI.Topology.CoarseAssembly.CountableAdditivity
-import OAI.Topology.CoarseAssembly.SeparatedRips
-import OAI.Topology.CoarseAssembly.RipsCompactSupport
-import OAI.Topology.CoarseAssembly.RipsFamilyMap
-import OAI.Topology.CoarseAssembly.RipsFamilyRestriction
-import OAI.Topology.CoarseAssembly.RipsPathCoarsening
-import OAI.Topology.CoarseAssembly.CocycleCircleMap
-import OAI.Topology.CoarseAssembly.RipsPaths
-import OAI.Topology.CoarseAssembly.CocyclePathLifting
-import OAI.Topology.CoarseAssembly.FiniteRips
-import OAI.Topology.CoarseAssembly.RipsWindingTest
-import OAI.Topology.CoarseAssembly.LoopCircle
-import OAI.Topology.CoarseAssembly.CirclePairing
-import OAI.Topology.CoarseAssembly.FiniteWindingIndex
-import OAI.Topology.CoarseAssembly.InclusionModule
-import OAI.Topology.CoarseAssembly.BoundaryEmbedding
-import OAI.Topology.CoarseAssembly.SeparatedUnion
-import OAI.Topology.CoarseAssembly.GraphMetric
-import OAI.Topology.CoarseAssembly.SeparatedGeometry
-import OAI.Topology.CoarseAssembly.ConcreteSpaces
-import OAI.Topology.CoarseAssembly.ConcreteTransferMaps
-import OAI.Topology.CoarseAssembly.ConcreteTransfer
-import OAI.Topology.CoarseAssembly.Averaging
-import OAI.Topology.CoarseAssembly.ProjectionBlocks
-import OAI.Topology.CoarseAssembly.SpatialAveraging
-import OAI.Topology.CoarseAssembly.AveragingPropagation
-import OAI.Topology.CoarseAssembly.KazhdanInput
-import OAI.Topology.CoarseAssembly.TensorFibers
-import OAI.Topology.CoarseAssembly.TransferCriterion
-import OAI.Topology.CoarseAssembly.FiniteFiberMaps
-import OAI.Topology.CoarseAssembly.CongruenceSurjective
-import OAI.Topology.CoarseAssembly.CongruenceEdgeOperators
-import OAI.Topology.CoarseAssembly.GraphFiberOperators
-import OAI.Topology.CoarseAssembly.PathFiberOperators
-import OAI.Topology.CoarseAssembly.DiscConnected
-import OAI.Topology.CoarseAssembly.FiberConnected
-import OAI.Topology.CoarseAssembly.FiberDegree
-import OAI.Topology.CoarseAssembly.AnalyticKHomology
-import OAI.Topology.CoarseAssembly.Assembly
-import OAI.Topology.CoarseAssembly.BoundaryIndex
-import OAI.Topology.CoarseAssembly.BlockOperators
-import OAI.Topology.CoarseAssembly.BoundedGeometry
-import OAI.Topology.CoarseAssembly.ChosenHolonomy
-import OAI.Topology.CoarseAssembly.CongruenceQuotients
-import OAI.Topology.CoarseAssembly.CongruenceMaps
-import OAI.Topology.CoarseAssembly.CongruenceFibers
-import OAI.Topology.CoarseAssembly.Conjugation
-import OAI.Topology.CoarseAssembly.ControlledMatrices
-import OAI.Topology.CoarseAssembly.FiniteWords
-import OAI.Topology.CoarseAssembly.FiberedGraph
-import OAI.Topology.CoarseAssembly.FiberConstants
-import OAI.Topology.CoarseAssembly.FiberEmbedding
-import OAI.Topology.CoarseAssembly.FredholmModule
-import OAI.Topology.CoarseAssembly.Holonomy
-import OAI.Topology.CoarseAssembly.HilbertSum
-import OAI.Topology.CoarseAssembly.Lattice
-import OAI.Topology.CoarseAssembly.LatticeCharge
-import OAI.Topology.CoarseAssembly.LocallyFinite
-import OAI.Topology.CoarseAssembly.NonUnitalK1
-import OAI.Topology.CoarseAssembly.Persistence
-import OAI.Topology.CoarseAssembly.RipsRealization
-import OAI.Topology.CoarseAssembly.Roe
-import OAI.Topology.CoarseAssembly.RoeCalculus
-import OAI.Topology.CoarseAssembly.RoeTransfer
-import OAI.Topology.CoarseAssembly.RoeRows
-import OAI.Topology.CoarseAssembly.Sandwich
-import OAI.Topology.CoarseAssembly.ShortLoops
-import OAI.Topology.CoarseAssembly.SpatialBlocks
-import OAI.Topology.CoarseAssembly.TopologicalK1
-import OAI.Topology.CoarseAssembly.TraceObstruction
-import OAI.Topology.CoarseAssembly.TraceSeparation
-import OAI.Topology.CoarseAssembly.TransferLift
-import OAI.Topology.CoarseAssembly.RealNormalization
-import OAI.Topology.CoarseAssembly.InvolutiveNormalization
-import OAI.Topology.CoarseAssembly.ScalarCycleNormalization
-import OAI.Topology.CoarseAssembly.LocalCompactPerturbation
-import OAI.Topology.CoarseAssembly.ScalarCutDown
-import OAI.Topology.CoarseAssembly.SupportedNormalization
-import OAI.Topology.CoarseAssembly.PointCoefficient
-import OAI.Topology.CoarseAssembly.FredholmParametrix
-import OAI.Topology.CoarseAssembly.ToeplitzCompression
-import OAI.Topology.CoarseAssembly.CycleIndex
-import OAI.Topology.CoarseAssembly.RepresentedCutDown
-import OAI.Topology.CoarseAssembly.RepresentedNormalization
-import OAI.Topology.CoarseAssembly.RepresentedIndex
-import OAI.Topology.CoarseAssembly.CompactIndex
-import OAI.Topology.CoarseAssembly.OperatorNormalization
-import OAI.Topology.CoarseAssembly.EssentialToeplitz
-import OAI.Topology.CoarseAssembly.ExplicitCutDown
-import OAI.Topology.CoarseAssembly.RawCycleIndex
-import OAI.Topology.CoarseAssembly.DegenerateCutDown
-import OAI.Topology.CoarseAssembly.UnitaryIndex
-import OAI.Topology.CoarseAssembly.CompactIndexPairing
-import OAI.Topology.CoarseAssembly.CircleInputReduction
-import OAI.Topology.CoarseAssembly.CircleCoefficient
-import OAI.Topology.CoarseAssembly.ExplicitCircleFamily
-import OAI.Topology.CoarseAssembly.ExplicitBoundaryConclusion
-import OAI.Topology.CoarseAssembly.StructureQuotient
-import OAI.Topology.CoarseAssembly.ControlledBoundaryHomotopy
-import OAI.Topology.CoarseAssembly.CayleyNaturality
-import OAI.Topology.CoarseAssembly.RipsPartition
-import OAI.Topology.CoarseAssembly.PartitionIsometry
-import OAI.Topology.CoarseAssembly.RipsBoundaryRepresentative
-import OAI.Topology.CoarseAssembly.AmpleRipsFamily
-import OAI.Topology.CoarseAssembly.MultiplierBoundaryHomotopy
-import OAI.Topology.CoarseAssembly.RelativeNaturality
-import OAI.Topology.CoarseAssembly.SeparatedFamilyHomotopy
-import OAI.Topology.CoarseAssembly.SeparatedFamilyPerturbation
-import OAI.Topology.CoarseAssembly.SpatialBoundaryNaturality
-import OAI.Topology.CoarseAssembly.CoefficientRoeHom
-import OAI.Topology.CoarseAssembly.PartitionInclusion
-import OAI.Topology.CoarseAssembly.CoefficientInclusion
-import OAI.Topology.CoarseAssembly.ExplicitBoundaryRepresentative
-import OAI.Topology.CoarseAssembly.NormalizedRipsFamily
-import OAI.Topology.CoarseAssembly.RipsBoundaryDegeneracy
-import OAI.Topology.CoarseAssembly.RipsFamilyCompactPerturbation
-import OAI.Topology.CoarseAssembly.RipsScaleCompatibility
-import OAI.Topology.CoarseAssembly.MultiplierUnitaryHomotopy
-import OAI.Topology.CoarseAssembly.RelativeTwoDiagonal
-import OAI.Topology.CoarseAssembly.IsometryDilation
-
-import OAI.Topology.CoarseAssembly.RelativePartialIsometry
-import OAI.Topology.CoarseAssembly.MultiplierIsometryInvariance
-
-import OAI.Topology.CoarseAssembly.RipsEmbeddingIndependence
-import OAI.Topology.CoarseAssembly.PointwiseEmbeddingTransition
-
-import OAI.Topology.CoarseAssembly.RipsUnitaryFamily
-import OAI.Topology.CoarseAssembly.OrthogonalSumBoundary
-
-import OAI.Topology.CoarseAssembly.RipsDirectSumBoundary
-
-import OAI.Topology.CoarseAssembly.SeparatedFamilyNearby
-
-import OAI.Topology.CoarseAssembly.ControlledCoarseAssembly
-import OAI.Topology.CoarseAssembly.RipsFiniteHomotopy
-import OAI.Topology.CoarseAssembly.RipsFixedRepresentationHomotopy
-
-import OAI.Topology.CoarseAssembly.RipsUnrestrictedHomotopy
-import OAI.Topology.CoarseAssembly.RegularizedRipsFamily
-import OAI.Topology.CoarseAssembly.BoundedTransformNaturality
-import OAI.Topology.CoarseAssembly.SeparatedRipsIndex
-import OAI.Topology.CoarseAssembly.RipsFiniteTail
-import OAI.Topology.CoarseAssembly.RegularizedCoarsening
-import OAI.Topology.CoarseAssembly.LateRipsCoarsening
-import OAI.Topology.CoarseAssembly.SeparatedCoarseAssembly
-import OAI.Topology.CoarseAssembly.SeparatedIndexNaturality
-import OAI.Topology.CoarseAssembly.IndexFamilyTransport
-import OAI.Topology.CoarseAssembly.ConstructedConclusion
-import OAI.Topology.CoarseAssembly.AveragingEnergy
-import OAI.Topology.CoarseAssembly.RegularConstants
-import OAI.Topology.CoarseAssembly.UnconditionalConclusion
-
-/-!
-# A bounded-geometry counterexample to coarse Novikov
-
-The concrete graph union has an infinite-order assembly-kernel class,
-and its constructed analytic assembly map fails to be injective both
-integrally and rationally.
--/
+import OAI.Topology.CoarseAssembly.FiberedPersistence
+import OAI.Topology.CoarseAssembly.HomotopyFilling
+import OAI.Topology.CoarseAssembly.LatticeHat
+import OAI.Topology.CoarseAssembly.FilledDisc
+import OAI.Topology.CoarseAssembly.FilledEdges
+import OAI.Topology.CoarseAssembly.FilledWalks
+import OAI.Topology.CoarseAssembly.CompactFamilyMap
+import OAI.Topology.CoarseAssembly.FamilyFilling
+import OAI.Topology.CoarseAssembly.RipsProperEmbedding
+import OAI.Topology.CoarseAssembly.RoeInclusion
+import OAI.Topology.CoarseAssembly.BoundaryGeometry
+import OAI.Topology.CoarseAssembly.BoundaryWalks
+import OAI.Topology.CoarseAssembly.BoundaryComponents
+import OAI.Topology.CoarseAssembly.HexagonSides
+import OAI.Topology.CoarseAssembly.RipsMapCalculus
+import OAI.Topology.CoarseAssembly.RipsReindex
+import OAI.Topology.CoarseAssembly.ConcretePersistence
+import OAI.Topology.CoarseAssembly.BoundaryLoopMaps
+import OAI.Topology.CoarseAssembly.BoundaryAssemblyMaps
+import OAI.Topology.CoarseAssembly.BoundaryConclusion
+import OAI.Topology.CoarseAssembly.PublishedInputReduction
+import OAI.Topology.CoarseAssembly.CycleCompactPerturbation
+import OAI.Topology.CoarseAssembly.UnitalCycleNormalization
+import OAI.Topology.CoarseAssembly.FiniteCycle
+import OAI.Topology.CoarseAssembly.FredholmSquare
+import OAI.Topology.CoarseAssembly.UnitalCycleInvertible
+import OAI.Topology.CoarseAssembly.CompactFunctionalCalculus
+import OAI.Topology.CoarseAssembly.RipsEmbedding
+import OAI.Topology.CoarseAssembly.ClopenRestriction
+import OAI.Topology.CoarseAssembly.ClopenNaturality
+import OAI.Topology.CoarseAssembly.CountableAdditivity
+import OAI.Topology.CoarseAssembly.SeparatedRips
+import OAI.Topology.CoarseAssembly.RipsCompactSupport
+import OAI.Topology.CoarseAssembly.RipsFamilyMap
+import OAI.Topology.CoarseAssembly.RipsFamilyRestriction
+import OAI.Topology.CoarseAssembly.RipsPathCoarsening
+import OAI.Topology.CoarseAssembly.CocycleCircleMap
+import OAI.Topology.CoarseAssembly.RipsPaths
+import OAI.Topology.CoarseAssembly.CocyclePathLifting
+import OAI.Topology.CoarseAssembly.FiniteRips
+import OAI.Topology.CoarseAssembly.RipsWindingTest
+import OAI.Topology.CoarseAssembly.LoopCircle
+import OAI.Topology.CoarseAssembly.CirclePairing
+import OAI.Topology.CoarseAssembly.FiniteWindingIndex
+import OAI.Topology.CoarseAssembly.InclusionModule
+import OAI.Topology.CoarseAssembly.BoundaryEmbedding
+import OAI.Topology.CoarseAssembly.SeparatedUnion
+import OAI.Topology.CoarseAssembly.GraphMetric
+import OAI.Topology.CoarseAssembly.SeparatedGeometry
+import OAI.Topology.CoarseAssembly.ConcreteSpaces
+import OAI.Topology.CoarseAssembly.ConcreteTransferMaps
+import OAI.Topology.CoarseAssembly.ConcreteTransfer
+import OAI.Topology.CoarseAssembly.Averaging
+import OAI.Topology.CoarseAssembly.ProjectionBlocks
+import OAI.Topology.CoarseAssembly.SpatialAveraging
+import OAI.Topology.CoarseAssembly.AveragingPropagation
+import OAI.Topology.CoarseAssembly.KazhdanInput
+import OAI.Topology.CoarseAssembly.TensorFibers
+import OAI.Topology.CoarseAssembly.TransferCriterion
+import OAI.Topology.CoarseAssembly.FiniteFiberMaps
+import OAI.Topology.CoarseAssembly.CongruenceSurjective
+import OAI.Topology.CoarseAssembly.CongruenceEdgeOperators
+import OAI.Topology.CoarseAssembly.GraphFiberOperators
+import OAI.Topology.CoarseAssembly.PathFiberOperators
+import OAI.Topology.CoarseAssembly.DiscConnected
+import OAI.Topology.CoarseAssembly.FiberConnected
+import OAI.Topology.CoarseAssembly.FiberDegree
+import OAI.Topology.CoarseAssembly.AnalyticKHomology
+import OAI.Topology.CoarseAssembly.Assembly
+import OAI.Topology.CoarseAssembly.BoundaryIndex
+import OAI.Topology.CoarseAssembly.BlockOperators
+import OAI.Topology.CoarseAssembly.BoundedGeometry
+import OAI.Topology.CoarseAssembly.ChosenHolonomy
+import OAI.Topology.CoarseAssembly.CongruenceQuotients
+import OAI.Topology.CoarseAssembly.CongruenceMaps
+import OAI.Topology.CoarseAssembly.CongruenceFibers
+import OAI.Topology.CoarseAssembly.Conjugation
+import OAI.Topology.CoarseAssembly.ControlledMatrices
+import OAI.Topology.CoarseAssembly.FiniteWords
+import OAI.Topology.CoarseAssembly.FiberedGraph
+import OAI.Topology.CoarseAssembly.FiberConstants
+import OAI.Topology.CoarseAssembly.FiberEmbedding
+import OAI.Topology.CoarseAssembly.FredholmModule
+import OAI.Topology.CoarseAssembly.Holonomy
+import OAI.Topology.CoarseAssembly.HilbertSum
+import OAI.Topology.CoarseAssembly.Lattice
+import OAI.Topology.CoarseAssembly.LatticeCharge
+import OAI.Topology.CoarseAssembly.LocallyFinite
+import OAI.Topology.CoarseAssembly.NonUnitalK1
+import OAI.Topology.CoarseAssembly.Persistence
+import OAI.Topology.CoarseAssembly.RipsRealization
+import OAI.Topology.CoarseAssembly.Roe
+import OAI.Topology.CoarseAssembly.RoeCalculus
+import OAI.Topology.CoarseAssembly.RoeTransfer
+import OAI.Topology.CoarseAssembly.RoeRows
+import OAI.Topology.CoarseAssembly.Sandwich
+import OAI.Topology.CoarseAssembly.ShortLoops
+import OAI.Topology.CoarseAssembly.SpatialBlocks
+import OAI.Topology.CoarseAssembly.TopologicalK1
+import OAI.Topology.CoarseAssembly.TraceObstruction
+import OAI.Topology.CoarseAssembly.TraceSeparation
+import OAI.Topology.CoarseAssembly.TransferLift
+import OAI.Topology.CoarseAssembly.RealNormalization
+import OAI.Topology.CoarseAssembly.InvolutiveNormalization
+import OAI.Topology.CoarseAssembly.ScalarCycleNormalization
+import OAI.Topology.CoarseAssembly.LocalCompactPerturbation
+import OAI.Topology.CoarseAssembly.ScalarCutDown
+import OAI.Topology.CoarseAssembly.SupportedNormalization
+import OAI.Topology.CoarseAssembly.PointCoefficient
+import OAI.Topology.CoarseAssembly.FredholmParametrix
+import OAI.Topology.CoarseAssembly.ToeplitzCompression
+import OAI.Topology.CoarseAssembly.CycleIndex
+import OAI.Topology.CoarseAssembly.RepresentedCutDown
+import OAI.Topology.CoarseAssembly.RepresentedNormalization
+import OAI.Topology.CoarseAssembly.RepresentedIndex
+import OAI.Topology.CoarseAssembly.CompactIndex
+import OAI.Topology.CoarseAssembly.OperatorNormalization
+import OAI.Topology.CoarseAssembly.EssentialToeplitz
+import OAI.Topology.CoarseAssembly.ExplicitCutDown
+import OAI.Topology.CoarseAssembly.RawCycleIndex
+import OAI.Topology.CoarseAssembly.DegenerateCutDown
+import OAI.Topology.CoarseAssembly.UnitaryIndex
+import OAI.Topology.CoarseAssembly.CompactIndexPairing
+import OAI.Topology.CoarseAssembly.CircleInputReduction
+import OAI.Topology.CoarseAssembly.CircleCoefficient
+import OAI.Topology.CoarseAssembly.ExplicitCircleFamily
+import OAI.Topology.CoarseAssembly.ExplicitBoundaryConclusion
+import OAI.Topology.CoarseAssembly.StructureQuotient
+import OAI.Topology.CoarseAssembly.ControlledBoundaryHomotopy
+import OAI.Topology.CoarseAssembly.CayleyNaturality
+import OAI.Topology.CoarseAssembly.RipsPartition
+import OAI.Topology.CoarseAssembly.PartitionIsometry
+import OAI.Topology.CoarseAssembly.RipsBoundaryRepresentative
+import OAI.Topology.CoarseAssembly.AmpleRipsFamily
+import OAI.Topology.CoarseAssembly.MultiplierBoundaryHomotopy
+import OAI.Topology.CoarseAssembly.RelativeNaturality
+import OAI.Topology.CoarseAssembly.SeparatedFamilyHomotopy
+import OAI.Topology.CoarseAssembly.SeparatedFamilyPerturbation
+import OAI.Topology.CoarseAssembly.SpatialBoundaryNaturality
+import OAI.Topology.CoarseAssembly.CoefficientRoeHom
+import OAI.Topology.CoarseAssembly.PartitionInclusion
+import OAI.Topology.CoarseAssembly.CoefficientInclusion
+import OAI.Topology.CoarseAssembly.ExplicitBoundaryRepresentative
+import OAI.Topology.CoarseAssembly.NormalizedRipsFamily
+import OAI.Topology.CoarseAssembly.RipsBoundaryDegeneracy
+import OAI.Topology.CoarseAssembly.RipsFamilyCompactPerturbation
+import OAI.Topology.CoarseAssembly.RipsScaleCompatibility
+import OAI.Topology.CoarseAssembly.MultiplierUnitaryHomotopy
+import OAI.Topology.CoarseAssembly.RelativeTwoDiagonal
+import OAI.Topology.CoarseAssembly.IsometryDilation
+
+import OAI.Topology.CoarseAssembly.RelativePartialIsometry
+import OAI.Topology.CoarseAssembly.MultiplierIsometryInvariance
+
+import OAI.Topology.CoarseAssembly.RipsEmbeddingIndependence
+import OAI.Topology.CoarseAssembly.PointwiseEmbeddingTransition
+
+import OAI.Topology.CoarseAssembly.RipsUnitaryFamily
+import OAI.Topology.CoarseAssembly.OrthogonalSumBoundary
+
+import OAI.Topology.CoarseAssembly.RipsDirectSumBoundary
+
+import OAI.Topology.CoarseAssembly.SeparatedFamilyNearby
+
+import OAI.Topology.CoarseAssembly.ControlledCoarseAssembly
+import OAI.Topology.CoarseAssembly.RipsFiniteHomotopy
+import OAI.Topology.CoarseAssembly.RipsFixedRepresentationHomotopy
+
+import OAI.Topology.CoarseAssembly.RipsUnrestrictedHomotopy
+import OAI.Topology.CoarseAssembly.RegularizedRipsFamily
+import OAI.Topology.CoarseAssembly.BoundedTransformNaturality
+import OAI.Topology.CoarseAssembly.SeparatedRipsIndex
+import OAI.Topology.CoarseAssembly.RipsFiniteTail
+import OAI.Topology.CoarseAssembly.RegularizedCoarsening
+import OAI.Topology.CoarseAssembly.LateRipsCoarsening
+import OAI.Topology.CoarseAssembly.SeparatedCoarseAssembly
+import OAI.Topology.CoarseAssembly.SeparatedIndexNaturality
+import OAI.Topology.CoarseAssembly.IndexFamilyTransport
+import OAI.Topology.CoarseAssembly.ConstructedConclusion
+import OAI.Topology.CoarseAssembly.AveragingEnergy
+import OAI.Topology.CoarseAssembly.RegularConstants
+import OAI.Topology.CoarseAssembly.UnconditionalConclusion
+
+/-!
+# A bounded-geometry counterexample to coarse Novikov
+
+The concrete graph union has an infinite-order assembly-kernel class,
+and its constructed analytic assembly map fails to be injective both
+integrally and rationally.
+-/

lean/OAI/Topology/CoarseAssembly/UnconditionalConclusion.lean

@@ -1,60 +1,60 @@
-import OAI.Topology.CoarseAssembly.Definitions
-import OAI.Topology.CoarseAssembly.CongruenceElementary
-import OAI.Topology.CoarseAssembly.ConstructedConclusion
-
-namespace OAI
-
-/-! # The unconditional bounded-geometry counterexample
-
-The geometric construction, analytic assembly, integral spectral certificate
-and finite congruence generation are combined here.
--/
-
-noncomputable section
-
-namespace CoarseAssembly
-
-open Congruence Rips Analytic
-
-attribute [local instance] OAI.CoarseAssembly.unconditionalConclusionMetric
-
-theorem provedKazhdanInput : PublishedKazhdanInput :=
-  publishedKazhdanInput_of_elementaryReduction elementaryReduction_surjective
-
-/-- The constructed ordinary coarse assembly map has an infinite-order
-kernel class and fails injectivity both integrally and rationally. -/
-theorem bounded_geometry_counterexample_unconditional :
-    ProperSpace VertexUnion ∧ UniformFiniteBalls VertexUnion ∧
-      (∀ x y : VertexUnion, x ≠ y → 1 ≤ dist x y) ∧
-      ∃ α : KX1 VertexUnion,
-        (∀ n : ℤ, n ≠ 0 → n • α ≠ 0) ∧ constructedCoarseAssembly α = 0 ∧
-        (1 : ℚ) ⊗ₜ[ℤ] α ≠ 0 ∧
-        ¬ Function.Injective constructedCoarseAssembly ∧
-        ¬ Function.Injective (constructedCoarseAssembly.toIntLinearMap.lTensor ℚ) :=
-  bounded_geometry_counterexample_from_gap provedKazhdanInput
-
-/-- The graph-family clauses of the main theorem, together with the full
-coarse-assembly conclusion for its concrete separated union. -/
-theorem bounded_geometry_graph_counterexample_unconditional :
-    (∀ j, Finite (FiberVertex j) ∧
-      (fiberedGraph elementaryMoves chosenHolonomyUnit j).Connected) ∧
-    (∃ D : ℕ, ∀ j (x : FiberVertex j),
-      Nat.card {y // (fiberedGraph elementaryMoves chosenHolonomyUnit j).Adj x y} ≤ D) ∧
-    ProperSpace VertexUnion ∧ UniformFiniteBalls VertexUnion ∧
-      (∀ x y : VertexUnion, x ≠ y → 1 ≤ dist x y) ∧
-      ∃ α : KX1 VertexUnion,
-        (∀ n : ℤ, n ≠ 0 → n • α ≠ 0) ∧ constructedCoarseAssembly α = 0 ∧
-        (1 : ℚ) ⊗ₜ[ℤ] α ≠ 0 ∧
-        ¬ Function.Injective constructedCoarseAssembly ∧
-        ¬ Function.Injective (constructedCoarseAssembly.toIntLinearMap.lTensor ℚ) := by
-  refine ⟨?_, ?_, bounded_geometry_counterexample_unconditional⟩
-  · intro j
-    exact ⟨inferInstance, fiberedGraph_connected elementaryMoves chosenHolonomyUnit j⟩
-  · exact ⟨7 * (symmetricMoves elementaryMoves).card * 2 ^ 9,
-      fun _ x => card_fibered_neighbors_le elementaryMoves chosenHolonomyUnit x⟩
-
-end CoarseAssembly
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.Definitions
+import OAI.Topology.CoarseAssembly.CongruenceElementary
+import OAI.Topology.CoarseAssembly.ConstructedConclusion
+
+namespace OAI
+
+/-! # The unconditional bounded-geometry counterexample
+
+The geometric construction, analytic assembly, integral spectral certificate
+and finite congruence generation are combined here.
+-/
+
+noncomputable section
+
+namespace CoarseAssembly
+
+open Congruence Rips Analytic
+
+attribute [local instance] OAI.CoarseAssembly.unconditionalConclusionMetric
+
+theorem provedKazhdanInput : PublishedKazhdanInput :=
+  publishedKazhdanInput_of_elementaryReduction elementaryReduction_surjective
+
+/-- The constructed ordinary coarse assembly map has an infinite-order
+kernel class and fails injectivity both integrally and rationally. -/
+theorem bounded_geometry_counterexample_unconditional :
+    ProperSpace VertexUnion ∧ UniformFiniteBalls VertexUnion ∧
+      (∀ x y : VertexUnion, x ≠ y → 1 ≤ dist x y) ∧
+      ∃ α : KX1 VertexUnion,
+        (∀ n : ℤ, n ≠ 0 → n • α ≠ 0) ∧ constructedCoarseAssembly α = 0 ∧
+        (1 : ℚ) ⊗ₜ[ℤ] α ≠ 0 ∧
+        ¬ Function.Injective constructedCoarseAssembly ∧
+        ¬ Function.Injective (constructedCoarseAssembly.toIntLinearMap.lTensor ℚ) :=
+  bounded_geometry_counterexample_from_gap provedKazhdanInput
+
+/-- The graph-family clauses of the main theorem, together with the full
+coarse-assembly conclusion for its concrete separated union. -/
+theorem bounded_geometry_graph_counterexample_unconditional :
+    (∀ j, Finite (FiberVertex j) ∧
+      (fiberedGraph elementaryMoves chosenHolonomyUnit j).Connected) ∧
+    (∃ D : ℕ, ∀ j (x : FiberVertex j),
+      Nat.card {y // (fiberedGraph elementaryMoves chosenHolonomyUnit j).Adj x y} ≤ D) ∧
+    ProperSpace VertexUnion ∧ UniformFiniteBalls VertexUnion ∧
+      (∀ x y : VertexUnion, x ≠ y → 1 ≤ dist x y) ∧
+      ∃ α : KX1 VertexUnion,
+        (∀ n : ℤ, n ≠ 0 → n • α ≠ 0) ∧ constructedCoarseAssembly α = 0 ∧
+        (1 : ℚ) ⊗ₜ[ℤ] α ≠ 0 ∧
+        ¬ Function.Injective constructedCoarseAssembly ∧
+        ¬ Function.Injective (constructedCoarseAssembly.toIntLinearMap.lTensor ℚ) := by
+  refine ⟨?_, ?_, bounded_geometry_counterexample_unconditional⟩
+  · intro j
+    exact ⟨inferInstance, fiberedGraph_connected elementaryMoves chosenHolonomyUnit j⟩
+  · exact ⟨7 * (symmetricMoves elementaryMoves).card * 2 ^ 9,
+      fun _ x => card_fibered_neighbors_le elementaryMoves chosenHolonomyUnit x⟩
+
+end CoarseAssembly
+
+end
+
+end OAI

lean/OAI/Topology/CoarseAssembly/UniformCycleNormalization.lean

@@ -1,53 +1,53 @@
-import OAI.Topology.CoarseAssembly.SigmaCycle
-import OAI.Topology.CoarseAssembly.RepresentedNormalization
-
-namespace OAI
-
-/-! # Uniformly bounded representatives for countable families -/
-
-noncomputable section
-
-namespace CoarseAssembly.Analytic
-
-open scoped ZeroAtInfty
-
-theorem involution_norm_le_one {H : HilbertSpace} (F : End H)
-    (hself : IsSelfAdjoint F) (hsq : F * F = 1) : ‖F‖ ≤ 1 := by
-  have he : ‖F‖ ^ 2 = ‖(1 : End H)‖ := by
-    rw [← hself.norm_mul_self, hsq]
-  have hi : ‖(1 : End H)‖ ≤ 1 := ContinuousLinearMap.norm_id_le
-  nlinarith [norm_nonneg F]
-
-variable {A : Type} [CStarAlgebra A]
-
-def normalizedCycle (c : OddCycle A) : OddCycle A :=
-  (exists_represented_involutive_representative c).choose
-
-theorem normalizedCycle_class (c : OddCycle A) :
-    cycleClass A (normalizedCycle c) = cycleClass A c :=
-  (exists_represented_involutive_representative c).choose_spec.1
-
-theorem normalizedCycle_selfAdjoint (c : OddCycle A) :
-    IsSelfAdjoint (normalizedCycle c).operator :=
-  (exists_represented_involutive_representative c).choose_spec.2.1
-
-theorem normalizedCycle_sq (c : OddCycle A) :
-    (normalizedCycle c).operator * (normalizedCycle c).operator = 1 :=
-  (exists_represented_involutive_representative c).choose_spec.2.2.1
-
-theorem normalizedCycle_norm (c : OddCycle A) : ‖(normalizedCycle c).operator‖ ≤ 1 :=
-  involution_norm_le_one _ (normalizedCycle_selfAdjoint c) (normalizedCycle_sq c)
-
-variable (Y : ℕ → Type) [∀ j, TopologicalSpace (Y j)] [∀ j, CompactSpace (Y j)]
-
-/-- Actual locally finite cycle for an arbitrary countable family of compact-source cycles. -/
-def normalizedFamilyCycle (c : ∀ j, OddCycle C(Y j, ℂ)) : OddCycle C₀(Sigma Y, ℂ) :=
-  sigmaCycle Y (fun j => normalizedCycle (c j)) 1 (by norm_num)
-    (fun j => normalizedCycle_norm (c j)) (fun j => normalizedCycle_selfAdjoint (c j))
-    (fun j => normalizedCycle_sq (c j))
-
-end CoarseAssembly.Analytic
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.SigmaCycle
+import OAI.Topology.CoarseAssembly.RepresentedNormalization
+
+namespace OAI
+
+/-! # Uniformly bounded representatives for countable families -/
+
+noncomputable section
+
+namespace CoarseAssembly.Analytic
+
+open scoped ZeroAtInfty
+
+theorem involution_norm_le_one {H : HilbertSpace} (F : End H)
+    (hself : IsSelfAdjoint F) (hsq : F * F = 1) : ‖F‖ ≤ 1 := by
+  have he : ‖F‖ ^ 2 = ‖(1 : End H)‖ := by
+    rw [← hself.norm_mul_self, hsq]
+  have hi : ‖(1 : End H)‖ ≤ 1 := ContinuousLinearMap.norm_id_le
+  nlinarith [norm_nonneg F]
+
+variable {A : Type} [CStarAlgebra A]
+
+def normalizedCycle (c : OddCycle A) : OddCycle A :=
+  (exists_represented_involutive_representative c).choose
+
+theorem normalizedCycle_class (c : OddCycle A) :
+    cycleClass A (normalizedCycle c) = cycleClass A c :=
+  (exists_represented_involutive_representative c).choose_spec.1
+
+theorem normalizedCycle_selfAdjoint (c : OddCycle A) :
+    IsSelfAdjoint (normalizedCycle c).operator :=
+  (exists_represented_involutive_representative c).choose_spec.2.1
+
+theorem normalizedCycle_sq (c : OddCycle A) :
+    (normalizedCycle c).operator * (normalizedCycle c).operator = 1 :=
+  (exists_represented_involutive_representative c).choose_spec.2.2.1
+
+theorem normalizedCycle_norm (c : OddCycle A) : ‖(normalizedCycle c).operator‖ ≤ 1 :=
+  involution_norm_le_one _ (normalizedCycle_selfAdjoint c) (normalizedCycle_sq c)
+
+variable (Y : ℕ → Type) [∀ j, TopologicalSpace (Y j)] [∀ j, CompactSpace (Y j)]
+
+/-- Actual locally finite cycle for an arbitrary countable family of compact-source cycles. -/
+def normalizedFamilyCycle (c : ∀ j, OddCycle C(Y j, ℂ)) : OddCycle C₀(Sigma Y, ℂ) :=
+  sigmaCycle Y (fun j => normalizedCycle (c j)) 1 (by norm_num)
+    (fun j => normalizedCycle_norm (c j)) (fun j => normalizedCycle_selfAdjoint (c j))
+    (fun j => normalizedCycle_sq (c j))
+
+end CoarseAssembly.Analytic
+
+end
+
+end OAI

lean/OAI/Topology/CoarseAssembly/UniformSubdivision.lean

@@ -1,47 +1,47 @@
-import OAI.Topology.CoarseAssembly.FredholmModule
-import Mathlib.Topology.UniformSpace.HeineCantor
-import Mathlib.Algebra.Order.Archimedean.Real.Basic
-
-namespace OAI
-
-/-! # Finite subdivisions of an actual norm-continuous homotopy -/
-
-noncomputable section
-
-namespace CoarseAssembly.Analytic
-
-def intervalGrid (N : ℕ) (hN : 0 < N) (j : ℕ) : Interval :=
-  ⟨((min j N : ℕ) : ℝ) / N, div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _),
-    (div_le_one (Nat.cast_pos.mpr hN)).mpr (by exact_mod_cast Nat.min_le_right j N)⟩
-
-@[simp] theorem intervalGrid_start (N : ℕ) (hN : 0 < N) : intervalGrid N hN 0 = start := by
-  apply Subtype.ext
-  simp [intervalGrid, start]
-
-@[simp] theorem intervalGrid_finish (N : ℕ) (hN : 0 < N) : intervalGrid N hN N = finish := by
-  apply Subtype.ext
-  simp [intervalGrid, finish, Nat.ne_of_gt hN]
-
-theorem intervalGrid_dist (N : ℕ) (hN : 0 < N) (j : ℕ) (hj : j < N) :
-    dist (intervalGrid N hN (j + 1)) (intervalGrid N hN j) = (N : ℝ)⁻¹ := by
-  change |((min (j + 1) N : ℕ) : ℝ) / N - ((min j N : ℕ) : ℝ) / N| = _
-  rw [Nat.min_eq_left (Nat.succ_le_of_lt hj), Nat.min_eq_left hj.le, Nat.cast_succ]
-  have he : ((j : ℝ) + 1) / N - (j : ℝ) / N = 1 / N := by ring
-  rw [he, abs_of_nonneg (by positivity), one_div]
-
-theorem exists_interval_subdivision {E : Type*} [PseudoMetricSpace E]
-    (f : Interval → E) (hf : Continuous f) (ε : ℝ) (hε : 0 < ε) :
-    ∃ (N : ℕ) (hN : 0 < N), ∀ j, j < N →
-      dist (f (intervalGrid N hN (j + 1))) (f (intervalGrid N hN j)) < ε := by
-  obtain ⟨δ, hδ, hclose⟩ := Metric.uniformContinuous_iff.mp
-    (CompactSpace.uniformContinuous_of_continuous hf) ε hε
-  obtain ⟨N, hN, hn⟩ := Real.exists_nat_pos_inv_lt hδ
-  refine ⟨N, hN, fun j hj => hclose ?_⟩
-  rw [intervalGrid_dist N hN j hj]
-  exact hn
-
-end CoarseAssembly.Analytic
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.FredholmModule
+import Mathlib.Topology.UniformSpace.HeineCantor
+import Mathlib.Algebra.Order.Archimedean.Real.Basic
+
+namespace OAI
+
+/-! # Finite subdivisions of an actual norm-continuous homotopy -/
+
+noncomputable section
+
+namespace CoarseAssembly.Analytic
+
+def intervalGrid (N : ℕ) (hN : 0 < N) (j : ℕ) : Interval :=
+  ⟨((min j N : ℕ) : ℝ) / N, div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _),
+    (div_le_one (Nat.cast_pos.mpr hN)).mpr (by exact_mod_cast Nat.min_le_right j N)⟩
+
+@[simp] theorem intervalGrid_start (N : ℕ) (hN : 0 < N) : intervalGrid N hN 0 = start := by
+  apply Subtype.ext
+  simp [intervalGrid, start]
+
+@[simp] theorem intervalGrid_finish (N : ℕ) (hN : 0 < N) : intervalGrid N hN N = finish := by
+  apply Subtype.ext
+  simp [intervalGrid, finish, Nat.ne_of_gt hN]
+
+theorem intervalGrid_dist (N : ℕ) (hN : 0 < N) (j : ℕ) (hj : j < N) :
+    dist (intervalGrid N hN (j + 1)) (intervalGrid N hN j) = (N : ℝ)⁻¹ := by
+  change |((min (j + 1) N : ℕ) : ℝ) / N - ((min j N : ℕ) : ℝ) / N| = _
+  rw [Nat.min_eq_left (Nat.succ_le_of_lt hj), Nat.min_eq_left hj.le, Nat.cast_succ]
+  have he : ((j : ℝ) + 1) / N - (j : ℝ) / N = 1 / N := by ring
+  rw [he, abs_of_nonneg (by positivity), one_div]
+
+theorem exists_interval_subdivision {E : Type*} [PseudoMetricSpace E]
+    (f : Interval → E) (hf : Continuous f) (ε : ℝ) (hε : 0 < ε) :
+    ∃ (N : ℕ) (hN : 0 < N), ∀ j, j < N →
+      dist (f (intervalGrid N hN (j + 1))) (f (intervalGrid N hN j)) < ε := by
+  obtain ⟨δ, hδ, hclose⟩ := Metric.uniformContinuous_iff.mp
+    (CompactSpace.uniformContinuous_of_continuous hf) ε hε
+  obtain ⟨N, hN, hn⟩ := Real.exists_nat_pos_inv_lt hδ
+  refine ⟨N, hN, fun j hj => hclose ?_⟩
+  rw [intervalGrid_dist N hN j hj]
+  exact hn
+
+end CoarseAssembly.Analytic
+
+end
+
+end OAI

lean/OAI/Topology/CoarseAssembly/UnitalCycleInvertible.lean

@@ -1,32 +1,32 @@
-import OAI.Topology.CoarseAssembly.FredholmInvertible
-
-namespace OAI
-
-/-! # Invertible self-adjoint representatives for unital odd cycles -/
-
-noncomputable section
-
-namespace CoarseAssembly.Analytic
-
-variable {A : Type} [CStarAlgebra A]
-
-theorem exists_invertible_selfAdjoint_representative (c : OddCycle A)
-    (hunital : c.representation 1 = 1) :
-    ∃ d : OddCycle A, cycleClass A d = cycleClass A c ∧
-      d.representation 1 = 1 ∧ IsSelfAdjoint d.operator ∧ IsUnit d.operator := by
-  obtain ⟨d, hdc, hdunit, hdself⟩ := exists_selfAdjoint_representative c hunital
-  have hc : IsCompactOperator (d.operator * d.operator - 1 : End d.space) := by
-    simpa only [hdunit, mul_one] using d.square_compact 1
-  have hp := kernelProjection_compact d.operator hc
-  have hps : IsCompactOperator (star (kernelProjection d.operator) : End d.space) := by
-    rw [(kernelProjection_selfAdjoint d.operator).star_eq]
-    exact hp
-  refine ⟨d.compactPerturb (kernelProjection d.operator) hp hps,
-    (compactPerturb_class d _ hp hps).trans hdc, hdunit,
-    kernel_corrected_selfAdjoint d.operator hdself, kernel_corrected_isUnit d.operator hdself hc⟩
-
-end CoarseAssembly.Analytic
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.FredholmInvertible
+
+namespace OAI
+
+/-! # Invertible self-adjoint representatives for unital odd cycles -/
+
+noncomputable section
+
+namespace CoarseAssembly.Analytic
+
+variable {A : Type} [CStarAlgebra A]
+
+theorem exists_invertible_selfAdjoint_representative (c : OddCycle A)
+    (hunital : c.representation 1 = 1) :
+    ∃ d : OddCycle A, cycleClass A d = cycleClass A c ∧
+      d.representation 1 = 1 ∧ IsSelfAdjoint d.operator ∧ IsUnit d.operator := by
+  obtain ⟨d, hdc, hdunit, hdself⟩ := exists_selfAdjoint_representative c hunital
+  have hc : IsCompactOperator (d.operator * d.operator - 1 : End d.space) := by
+    simpa only [hdunit, mul_one] using d.square_compact 1
+  have hp := kernelProjection_compact d.operator hc
+  have hps : IsCompactOperator (star (kernelProjection d.operator) : End d.space) := by
+    rw [(kernelProjection_selfAdjoint d.operator).star_eq]
+    exact hp
+  refine ⟨d.compactPerturb (kernelProjection d.operator) hp hps,
+    (compactPerturb_class d _ hp hps).trans hdc, hdunit,
+    kernel_corrected_selfAdjoint d.operator hdself, kernel_corrected_isUnit d.operator hdself hc⟩
+
+end CoarseAssembly.Analytic
+
+end
+
+end OAI

lean/OAI/Topology/CoarseAssembly/UnitalCycleNormalization.lean

@@ -1,43 +1,43 @@
-import OAI.Topology.CoarseAssembly.CycleCompactPerturbation
-import Mathlib.Tactic.Module
-
-namespace OAI
-
-/-! # Self-adjoint normalization of a unital Fredholm cycle
-
-The compact adjoint defect allows replacement of the operator by its
-self-adjoint part, with the represented algebra and K-homology class fixed.
--/
-
-noncomputable section
-
-namespace CoarseAssembly.Analytic
-
-variable {A : Type} [CStarAlgebra A]
-
-theorem exists_selfAdjoint_representative (c : OddCycle A)
-    (hunital : c.representation 1 = 1) :
-    ∃ d : OddCycle A, cycleClass A d = cycleClass A c ∧
-      d.representation 1 = 1 ∧ IsSelfAdjoint d.operator := by
-  let K : End c.space := (1 / 2 : ℝ) • (star c.operator - c.operator)
-  have hdefect : IsCompactOperator (star c.operator - c.operator : End c.space) := by
-    simpa only [hunital, mul_one] using c.adjoint_compact 1
-  have hK : IsCompactOperator K := hdefect.smul (1 / 2 : ℝ)
-  have hstar : star K = -K := by
-    dsimp only [K]
-    rw [star_smul, star_trivial, star_sub, star_star]
-    module
-  have hKs : IsCompactOperator (star K : End c.space) := by
-    rw [hstar]
-    exact hK.neg
-  refine ⟨c.compactPerturb K hK hKs, compactPerturb_class c K hK hKs, hunital, ?_⟩
-  change star (c.operator + K) = c.operator + K
-  rw [star_add, hstar]
-  dsimp only [K]
-  module
-
-end CoarseAssembly.Analytic
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.CycleCompactPerturbation
+import Mathlib.Tactic.Module
+
+namespace OAI
+
+/-! # Self-adjoint normalization of a unital Fredholm cycle
+
+The compact adjoint defect allows replacement of the operator by its
+self-adjoint part, with the represented algebra and K-homology class fixed.
+-/
+
+noncomputable section
+
+namespace CoarseAssembly.Analytic
+
+variable {A : Type} [CStarAlgebra A]
+
+theorem exists_selfAdjoint_representative (c : OddCycle A)
+    (hunital : c.representation 1 = 1) :
+    ∃ d : OddCycle A, cycleClass A d = cycleClass A c ∧
+      d.representation 1 = 1 ∧ IsSelfAdjoint d.operator := by
+  let K : End c.space := (1 / 2 : ℝ) • (star c.operator - c.operator)
+  have hdefect : IsCompactOperator (star c.operator - c.operator : End c.space) := by
+    simpa only [hunital, mul_one] using c.adjoint_compact 1
+  have hK : IsCompactOperator K := hdefect.smul (1 / 2 : ℝ)
+  have hstar : star K = -K := by
+    dsimp only [K]
+    rw [star_smul, star_trivial, star_sub, star_star]
+    module
+  have hKs : IsCompactOperator (star K : End c.space) := by
+    rw [hstar]
+    exact hK.neg
+  refine ⟨c.compactPerturb K hK hKs, compactPerturb_class c K hK hKs, hunital, ?_⟩
+  change star (c.operator + K) = c.operator + K
+  rw [star_add, hstar]
+  dsimp only [K]
+  module
+
+end CoarseAssembly.Analytic
+
+end
+
+end OAI

lean/OAI/Topology/CoarseAssembly/UnitalRepresentative.lean

@@ -1,130 +1,130 @@
-import OAI.Topology.CoarseAssembly.BinaryCycle
-import OAI.Topology.CoarseAssembly.CharacterExtension
-import OAI.Topology.CoarseAssembly.ProjectionSwap
-import OAI.Topology.CoarseAssembly.UniformCycleNormalization
-
-namespace OAI
-
-/-! # Unital representatives for compact-source analytic classes
-
-Adjoin a degenerate character cycle and exchange the complementary
-subspaces by the explicit projection-swap unitary. This proves that
-completion of a representation by a character preserves its class.
--/
-
-noncomputable section
-
-namespace CoarseAssembly.Analytic
-
-variable {H : HilbertSpace}
-
-theorem projectionSwap_representation (P R : End H) (hP : IsStarProjection P)
-    (hPR : P * R = R) (hRP : R * P = R) (z : ℂ) :
-    binaryBlock H H (R + z • (1 - P)) (z • P) =
-      operatorConjugation (projectionSwapUnitary H P hP).toLinearIsometry
-        (binaryBlock H H R (z • 1)) := by
-  rw [projectionSwapUnitary_conjugation]
-  have hu := projectionSwap_sq H P hP.isIdempotentElem
-  suffices hi : binaryBlock H H (R + z • (1 - P)) (z • P) * projectionSwap H P =
-      projectionSwap H P * binaryBlock H H R (z • 1) by
-    have hi' := congrArg (fun T => T * projectionSwap H P) hi
-    simpa only [mul_assoc, hu, mul_one] using hi'
-  apply ContinuousLinearMap.ext
-  intro x
-  change binaryBlock H H (R + z • (1 - P)) (z • P) (projectionSwap H P x) =
-    projectionSwap H P (binaryBlock H H R (z • 1) x)
-  rw [projectionSwap_apply H, binaryBlock_apply H H, binaryBlock_apply H H,
-    projectionSwap_apply H]
-  apply (WithLp.ofLp_injective 2)
-  have hp (y : H) : P (P y) = P y := DFunLike.congr_fun hP.isIdempotentElem y
-  have hpr (y : H) : P (R y) = R y := DFunLike.congr_fun hPR y
-  have hrp (y : H) : R (P y) = R y := DFunLike.congr_fun hRP y
-  change
-    ((R + z • (1 - P)) (P x.fst + (1 - P) x.snd),
-      (z • P) ((1 - P) x.fst + P x.snd)) =
-    (P (R x.fst) + (1 - P) ((z • (1 : End H)) x.snd),
-      (1 - P) (R x.fst) + P ((z • (1 : End H)) x.snd))
-  simp only [add_apply, smul_apply, sub_apply, one_apply_eq_self, map_add, map_sub,
-    map_smul, hp, hpr, hrp]
-  ext <;> module
-
-theorem projectionSwap_operator (P F : End H) (hP : IsStarProjection P)
-    (hF : IsSelfAdjoint F) (hQ : F * (1 - P) = 1 - P) :
-    binaryBlock H H F 1 =
-      operatorConjugation (projectionSwapUnitary H P hP).toLinearIsometry
-        (binaryBlock H H F 1) := by
-  rw [projectionSwapUnitary_conjugation]
-  have hu := projectionSwap_sq H P hP.isIdempotentElem
-  have hc := (commute_of_complement F P hF hP.isSelfAdjoint hQ).eq
-  have hQ' : (1 - P) * F = 1 - P := by
-    have h := congrArg star hQ
-    simpa only [star_mul, star_sub, star_one, hP.isSelfAdjoint.star_eq, hF.star_eq] using h
-  suffices hi : binaryBlock H H F 1 * projectionSwap H P =
-      projectionSwap H P * binaryBlock H H F 1 by
-    have hi' := congrArg (fun T => T * projectionSwap H P) hi
-    simpa only [mul_assoc, hu, mul_one] using hi'
-  apply ContinuousLinearMap.ext
-  intro x
-  change binaryBlock H H F 1 (projectionSwap H P x) =
-    projectionSwap H P (binaryBlock H H F 1 x)
-  rw [projectionSwap_apply H, binaryBlock_apply H H, binaryBlock_apply H H,
-    projectionSwap_apply H]
-  apply (WithLp.ofLp_injective 2)
-  have hfp (y : H) : F (P y) = P (F y) := DFunLike.congr_fun hc y
-  have hfq (y : H) : F ((1 - P) y) = (1 - P) y := DFunLike.congr_fun hQ y
-  have hqf (y : H) : (1 - P) (F y) = (1 - P) y := DFunLike.congr_fun hQ' y
-  change (F (P x.fst + (1 - P) x.snd), (1 - P) x.fst + P x.snd) =
-    (P (F x.fst) + (1 - P) x.snd, (1 - P) (F x.fst) + P x.snd)
-  rw [map_add, hfp, hfq, hqf]
-
-variable {A : Type} [CStarAlgebra A]
-
-theorem characterExtension_class (c : OddCycle A) (χ : A →⋆ₐ[ℂ] ℂ)
-    (hF : IsSelfAdjoint c.operator)
-    (hQ : c.operator * (1 - c.representation 1) = 1 - c.representation 1)
-    (hsq : IsCompactOperator (c.operator * c.operator - 1 : End c.space)) :
-    cycleClass A (c.characterExtension χ hF hQ hsq) = cycleClass A c := by
-  let d := identityOperatorCycle
-    (characterProjectionRepresentation χ (1 : End c.space) (IsStarProjection.one _))
-  let e := identityOperatorCycle
-    (characterProjectionRepresentation χ (c.representation 1) (representation_unit_projection c))
-  have he : (c.binarySum d).UnitaryEquivalent
-      ((c.characterExtension χ hF hQ hsq).binarySum e) := by
-    refine ⟨projectionSwapUnitary c.space (c.representation 1) (representation_unit_projection c),
-      ?_, ?_⟩
-    · intro a
-      change binaryBlock c.space c.space
-        (c.representation a + χ a • (1 - c.representation 1)) (χ a • c.representation 1) = _
-      exact projectionSwap_representation _ _ (representation_unit_projection c)
-        (by rw [← map_mul, one_mul]) (by rw [← map_mul, mul_one]) (χ a)
-    · exact projectionSwap_operator _ _ (representation_unit_projection c) hF hQ
-  have h := unitary_eq _ _ he
-  rw [binarySum_class, binarySum_class,
-    degenerate_zero d (identityOperatorCycle_degenerate _),
-    degenerate_zero e (identityOperatorCycle_degenerate _), add_zero, add_zero] at h
-  exact h.symm
-
-/-- Every analytic cycle over an algebra with a character has a unital
-representative. The assertion includes equality in the actual cycle quotient. -/
-theorem exists_unital_representative (c : OddCycle A) (χ : A →⋆ₐ[ℂ] ℂ) :
-    ∃ d : OddCycle A, cycleClass A d = cycleClass A c ∧ d.representation 1 = 1 := by
-  let d := c.cutDown
-  let e := d.characterExtension χ (cutDown_selfAdjoint c) (cutDown_complement c)
-    (cutDown_square_compact c)
-  refine ⟨e, ?_, characterExtension_unital d χ _ _ _⟩
-  exact (characterExtension_class d χ _ _ _).trans (cutDown_class c)
-
-/-- The representatives needed by whole-family Rips localization can be
-chosen unital, self-adjoint, involutive, and uniformly bounded by one. -/
-theorem exists_unital_involutive_representative (c : OddCycle A) (χ : A →⋆ₐ[ℂ] ℂ) :
-    ∃ d : OddCycle A, cycleClass A d = cycleClass A c ∧ d.representation 1 = 1 ∧
-      IsSelfAdjoint d.operator ∧ d.operator * d.operator = 1 ∧ ‖d.operator‖ ≤ 1 := by
-  obtain ⟨d, hdc, hdu⟩ := exists_unital_representative c χ
-  obtain ⟨e, hed, heu, hes, heq⟩ := exists_involutive_representative d hdu
-  exact ⟨e, hed.trans hdc, heu, hes, heq, involution_norm_le_one _ hes heq⟩
-
-end CoarseAssembly.Analytic
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.BinaryCycle
+import OAI.Topology.CoarseAssembly.CharacterExtension
+import OAI.Topology.CoarseAssembly.ProjectionSwap
+import OAI.Topology.CoarseAssembly.UniformCycleNormalization
+
+namespace OAI
+
+/-! # Unital representatives for compact-source analytic classes
+
+Adjoin a degenerate character cycle and exchange the complementary
+subspaces by the explicit projection-swap unitary. This proves that
+completion of a representation by a character preserves its class.
+-/
+
+noncomputable section
+
+namespace CoarseAssembly.Analytic
+
+variable {H : HilbertSpace}
+
+theorem projectionSwap_representation (P R : End H) (hP : IsStarProjection P)
+    (hPR : P * R = R) (hRP : R * P = R) (z : ℂ) :
+    binaryBlock H H (R + z • (1 - P)) (z • P) =
+      operatorConjugation (projectionSwapUnitary H P hP).toLinearIsometry
+        (binaryBlock H H R (z • 1)) := by
+  rw [projectionSwapUnitary_conjugation]
+  have hu := projectionSwap_sq H P hP.isIdempotentElem
+  suffices hi : binaryBlock H H (R + z • (1 - P)) (z • P) * projectionSwap H P =
+      projectionSwap H P * binaryBlock H H R (z • 1) by
+    have hi' := congrArg (fun T => T * projectionSwap H P) hi
+    simpa only [mul_assoc, hu, mul_one] using hi'
+  apply ContinuousLinearMap.ext
+  intro x
+  change binaryBlock H H (R + z • (1 - P)) (z • P) (projectionSwap H P x) =
+    projectionSwap H P (binaryBlock H H R (z • 1) x)
+  rw [projectionSwap_apply H, binaryBlock_apply H H, binaryBlock_apply H H,
+    projectionSwap_apply H]
+  apply (WithLp.ofLp_injective 2)
+  have hp (y : H) : P (P y) = P y := DFunLike.congr_fun hP.isIdempotentElem y
+  have hpr (y : H) : P (R y) = R y := DFunLike.congr_fun hPR y
+  have hrp (y : H) : R (P y) = R y := DFunLike.congr_fun hRP y
+  change
+    ((R + z • (1 - P)) (P x.fst + (1 - P) x.snd),
+      (z • P) ((1 - P) x.fst + P x.snd)) =
+    (P (R x.fst) + (1 - P) ((z • (1 : End H)) x.snd),
+      (1 - P) (R x.fst) + P ((z • (1 : End H)) x.snd))
+  simp only [add_apply, smul_apply, sub_apply, one_apply_eq_self, map_add, map_sub,
+    map_smul, hp, hpr, hrp]
+  ext <;> module
+
+theorem projectionSwap_operator (P F : End H) (hP : IsStarProjection P)
+    (hF : IsSelfAdjoint F) (hQ : F * (1 - P) = 1 - P) :
+    binaryBlock H H F 1 =
+      operatorConjugation (projectionSwapUnitary H P hP).toLinearIsometry
+        (binaryBlock H H F 1) := by
+  rw [projectionSwapUnitary_conjugation]
+  have hu := projectionSwap_sq H P hP.isIdempotentElem
+  have hc := (commute_of_complement F P hF hP.isSelfAdjoint hQ).eq
+  have hQ' : (1 - P) * F = 1 - P := by
+    have h := congrArg star hQ
+    simpa only [star_mul, star_sub, star_one, hP.isSelfAdjoint.star_eq, hF.star_eq] using h
+  suffices hi : binaryBlock H H F 1 * projectionSwap H P =
+      projectionSwap H P * binaryBlock H H F 1 by
+    have hi' := congrArg (fun T => T * projectionSwap H P) hi
+    simpa only [mul_assoc, hu, mul_one] using hi'
+  apply ContinuousLinearMap.ext
+  intro x
+  change binaryBlock H H F 1 (projectionSwap H P x) =
+    projectionSwap H P (binaryBlock H H F 1 x)
+  rw [projectionSwap_apply H, binaryBlock_apply H H, binaryBlock_apply H H,
+    projectionSwap_apply H]
+  apply (WithLp.ofLp_injective 2)
+  have hfp (y : H) : F (P y) = P (F y) := DFunLike.congr_fun hc y
+  have hfq (y : H) : F ((1 - P) y) = (1 - P) y := DFunLike.congr_fun hQ y
+  have hqf (y : H) : (1 - P) (F y) = (1 - P) y := DFunLike.congr_fun hQ' y
+  change (F (P x.fst + (1 - P) x.snd), (1 - P) x.fst + P x.snd) =
+    (P (F x.fst) + (1 - P) x.snd, (1 - P) (F x.fst) + P x.snd)
+  rw [map_add, hfp, hfq, hqf]
+
+variable {A : Type} [CStarAlgebra A]
+
+theorem characterExtension_class (c : OddCycle A) (χ : A →⋆ₐ[ℂ] ℂ)
+    (hF : IsSelfAdjoint c.operator)
+    (hQ : c.operator * (1 - c.representation 1) = 1 - c.representation 1)
+    (hsq : IsCompactOperator (c.operator * c.operator - 1 : End c.space)) :
+    cycleClass A (c.characterExtension χ hF hQ hsq) = cycleClass A c := by
+  let d := identityOperatorCycle
+    (characterProjectionRepresentation χ (1 : End c.space) (IsStarProjection.one _))
+  let e := identityOperatorCycle
+    (characterProjectionRepresentation χ (c.representation 1) (representation_unit_projection c))
+  have he : (c.binarySum d).UnitaryEquivalent
+      ((c.characterExtension χ hF hQ hsq).binarySum e) := by
+    refine ⟨projectionSwapUnitary c.space (c.representation 1) (representation_unit_projection c),
+      ?_, ?_⟩
+    · intro a
+      change binaryBlock c.space c.space
+        (c.representation a + χ a • (1 - c.representation 1)) (χ a • c.representation 1) = _
+      exact projectionSwap_representation _ _ (representation_unit_projection c)
+        (by rw [← map_mul, one_mul]) (by rw [← map_mul, mul_one]) (χ a)
+    · exact projectionSwap_operator _ _ (representation_unit_projection c) hF hQ
+  have h := unitary_eq _ _ he
+  rw [binarySum_class, binarySum_class,
+    degenerate_zero d (identityOperatorCycle_degenerate _),
+    degenerate_zero e (identityOperatorCycle_degenerate _), add_zero, add_zero] at h
+  exact h.symm
+
+/-- Every analytic cycle over an algebra with a character has a unital
+representative. The assertion includes equality in the actual cycle quotient. -/
+theorem exists_unital_representative (c : OddCycle A) (χ : A →⋆ₐ[ℂ] ℂ) :
+    ∃ d : OddCycle A, cycleClass A d = cycleClass A c ∧ d.representation 1 = 1 := by
+  let d := c.cutDown
+  let e := d.characterExtension χ (cutDown_selfAdjoint c) (cutDown_complement c)
+    (cutDown_square_compact c)
+  refine ⟨e, ?_, characterExtension_unital d χ _ _ _⟩
+  exact (characterExtension_class d χ _ _ _).trans (cutDown_class c)
+
+/-- The representatives needed by whole-family Rips localization can be
+chosen unital, self-adjoint, involutive, and uniformly bounded by one. -/
+theorem exists_unital_involutive_representative (c : OddCycle A) (χ : A →⋆ₐ[ℂ] ℂ) :
+    ∃ d : OddCycle A, cycleClass A d = cycleClass A c ∧ d.representation 1 = 1 ∧
+      IsSelfAdjoint d.operator ∧ d.operator * d.operator = 1 ∧ ‖d.operator‖ ≤ 1 := by
+  obtain ⟨d, hdc, hdu⟩ := exists_unital_representative c χ
+  obtain ⟨e, hed, heu, hes, heq⟩ := exists_involutive_representative d hdu
+  exact ⟨e, hed.trans hdc, heu, hes, heq, involution_norm_le_one _ hes heq⟩
+
+end CoarseAssembly.Analytic
+
+end
+
+end OAI

lean/OAI/Topology/CoarseAssembly/UnitalStructureAlgebra.lean

@@ -1,66 +1,66 @@
-import OAI.Topology.CoarseAssembly.StructureQuotient
-import Mathlib.Analysis.CStarAlgebra.Spectrum
-
-namespace OAI
-
-/-! # The unit of the controlled pseudolocal algebra -/
-
-noncomputable section
-
-namespace CoarseAssembly.DiscreteRoe
-
-variable {X H : Type*} [DecidableEq X] [PseudoMetricSpace X]
-  [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
-
-local instance : IsScalarTower ℂ (Operator X H) (Operator X H) where
-  smul_assoc c T S := by ext v; simp
-local instance : SMulCommClass ℂ (Operator X H) (Operator X H) where
-  smul_comm c T S := by ext v; simp
-local instance : ContinuousStar (Operator X H) := ⟨operator_star_continuous⟩
-
-omit [CompleteSpace H] in
-theorem propagationLE_one : PropagationLE 0 (1 : Operator X H) := by
-  intro x y hxy
-  have hne : x ≠ y := by intro h; simp [h] at hxy
-  ext v
-  simp [entry_apply, lp.single_apply, hne]
-
-theorem structureGenerator_one : StructureGenerator (1 : Operator X H) := by
-  refine ⟨⟨0, le_rfl, propagationLE_one⟩, ?_, ?_⟩
-  · intro f
-    simp only [one_mul, mul_one, sub_self]
-    exact isCompactOperator_zero
-  · rw [star_one (Operator X H)]
-    intro f
-    simp only [one_mul, mul_one, sub_self]
-    exact isCompactOperator_zero
-
-theorem structureAlgebra_one : (1 : Operator X H) ∈ structureAlgebra (X := X) (H := H) :=
-  (NonUnitalStarSubalgebra.le_topologicalClosure _)
-    (NonUnitalStarAlgebra.subset_adjoin ℂ _ (structureGenerator_one (X := X) (H := H)))
-
-/-- The controlled structure algebra with its actual identity operator. -/
-def unitalStructureAlgebra : StarSubalgebra ℂ (Operator X H) where
-  carrier := structureAlgebra (X := X) (H := H)
-  add_mem' := (structureAlgebra (X := X) (H := H)).add_mem
-  mul_mem' := (structureAlgebra (X := X) (H := H)).mul_mem
-  star_mem' := fun h => star_mem h
-  algebraMap_mem' z := by
-    rw [Algebra.algebraMap_eq_smul_one]
-    exact (structureAlgebra (X := X) (H := H)).smul_mem z structureAlgebra_one
-
-theorem unitalStructureAlgebra_closed :
-    IsClosed (unitalStructureAlgebra (X := X) (H := H) : Set (Operator X H)) :=
-  NonUnitalStarSubalgebra.isClosed_topologicalClosure _
-
-instance : CompleteSpace (unitalStructureAlgebra (X := X) (H := H)) :=
-  unitalStructureAlgebra_closed.completeSpace_coe
-
-instance : CStarAlgebra (unitalStructureAlgebra (X := X) (H := H)) :=
-  StarSubalgebra.cstarAlgebra _ (h_closed := unitalStructureAlgebra_closed)
-
-end CoarseAssembly.DiscreteRoe
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.StructureQuotient
+import Mathlib.Analysis.CStarAlgebra.Spectrum
+
+namespace OAI
+
+/-! # The unit of the controlled pseudolocal algebra -/
+
+noncomputable section
+
+namespace CoarseAssembly.DiscreteRoe
+
+variable {X H : Type*} [DecidableEq X] [PseudoMetricSpace X]
+  [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
+
+local instance : IsScalarTower ℂ (Operator X H) (Operator X H) where
+  smul_assoc c T S := by ext v; simp
+local instance : SMulCommClass ℂ (Operator X H) (Operator X H) where
+  smul_comm c T S := by ext v; simp
+local instance : ContinuousStar (Operator X H) := ⟨operator_star_continuous⟩
+
+omit [CompleteSpace H] in
+theorem propagationLE_one : PropagationLE 0 (1 : Operator X H) := by
+  intro x y hxy
+  have hne : x ≠ y := by intro h; simp [h] at hxy
+  ext v
+  simp [entry_apply, lp.single_apply, hne]
+
+theorem structureGenerator_one : StructureGenerator (1 : Operator X H) := by
+  refine ⟨⟨0, le_rfl, propagationLE_one⟩, ?_, ?_⟩
+  · intro f
+    simp only [one_mul, mul_one, sub_self]
+    exact isCompactOperator_zero
+  · rw [star_one (Operator X H)]
+    intro f
+    simp only [one_mul, mul_one, sub_self]
+    exact isCompactOperator_zero
+
+theorem structureAlgebra_one : (1 : Operator X H) ∈ structureAlgebra (X := X) (H := H) :=
+  (NonUnitalStarSubalgebra.le_topologicalClosure _)
+    (NonUnitalStarAlgebra.subset_adjoin ℂ _ (structureGenerator_one (X := X) (H := H)))
+
+/-- The controlled structure algebra with its actual identity operator. -/
+def unitalStructureAlgebra : StarSubalgebra ℂ (Operator X H) where
+  carrier := structureAlgebra (X := X) (H := H)
+  add_mem' := (structureAlgebra (X := X) (H := H)).add_mem
+  mul_mem' := (structureAlgebra (X := X) (H := H)).mul_mem
+  star_mem' := fun h => star_mem h
+  algebraMap_mem' z := by
+    rw [Algebra.algebraMap_eq_smul_one]
+    exact (structureAlgebra (X := X) (H := H)).smul_mem z structureAlgebra_one
+
+theorem unitalStructureAlgebra_closed :
+    IsClosed (unitalStructureAlgebra (X := X) (H := H) : Set (Operator X H)) :=
+  NonUnitalStarSubalgebra.isClosed_topologicalClosure _
+
+instance : CompleteSpace (unitalStructureAlgebra (X := X) (H := H)) :=
+  unitalStructureAlgebra_closed.completeSpace_coe
+
+instance : CStarAlgebra (unitalStructureAlgebra (X := X) (H := H)) :=
+  StarSubalgebra.cstarAlgebra _ (h_closed := unitalStructureAlgebra_closed)
+
+end CoarseAssembly.DiscreteRoe
+
+end
+
+end OAI

lean/OAI/Topology/CoarseAssembly/UnitaryHomotopyIndex.lean

@@ -1,59 +1,59 @@
-import OAI.Topology.CoarseAssembly.CompactIndexPairing
-
-namespace OAI
-
-/-! # Homotopy invariance in the unitary argument of the index pairing -/
-
-noncomputable section
-
-namespace CoarseAssembly.Analytic
-
-open scoped CStarAlgebra
-
-variable {A : Type} [CStarAlgebra A]
-
-theorem rawCycleIndex_unitaryHomotopy (c : OddCycle A) (u : Interval → A)
-    (hu : Continuous u) (hul : ∀ t, star (u t) * u t = 1)
-    (hur : ∀ t, u t * star (u t) = 1) :
-    rawCycleIndex c (u start) = rawCycleIndex c (u finish) := by
-  apply fredholmIndex_homotopy (fun t => rawCycleToeplitz c (u t))
-  · have hrep : Continuous (fun t => c.representation (u t)) := (map_continuous c.representation).comp hu
-    have hext : Continuous (fun t => unitaryExtension c (u t)) := hrep.add continuous_const
-    exact ((continuous_const.mul hext).mul continuous_const).add continuous_const
-  · intro t
-    exact rawCycleToeplitz_fredholm c (u t) (hul t) (hur t)
-
-theorem indexPairing_unitaryHomotopy (u : Interval → A) (hu : Continuous u)
-    (hul : ∀ t, star (u t) * u t = 1) (hur : ∀ t, u t * star (u t) = 1) :
-    indexPairing (u start) (hul start) (hur start) =
-      indexPairing (u finish) (hul finish) (hur finish) := by
-  apply hom_ext
-  intro c
-  simp only [indexPairing_cycleClass]
-  exact rawCycleIndex_unitaryHomotopy c u hu hul hur
-
-open scoped ZeroAtInfty
-
-variable {X : Type} [TopologicalSpace X] [CompactSpace X]
-
-theorem compactIndexPairing_homotopy {f g : C(X, Circle)} (F : f.Homotopy g) :
-    compactIndexPairing f = compactIndexPairing g := by
-  let u : Interval → C(X, ℂ) := fun t => compactUnitary (homotopySlice F t)
-  have hu : Continuous u := by
-    let coeCircle : C(Circle, ℂ) := ⟨fun z => (z : ℂ), continuous_subtype_val⟩
-    exact (ContinuousMap.continuous_postcomp coeCircle).comp F.toContinuousMap.curry.continuous
-  have hul (t : Interval) : star (u t) * u t = 1 := compactUnitary_star_mul _
-  have hur (t : Interval) : u t * star (u t) = 1 := compactUnitary_mul_star _
-  have hi := indexPairing_unitaryHomotopy u hu hul hur
-  have he : indexPairing (compactUnitary f) (compactUnitary_star_mul f)
-      (compactUnitary_mul_star f) = indexPairing (compactUnitary g)
-      (compactUnitary_star_mul g) (compactUnitary_mul_star g) := by
-    simpa only [u, homotopySlice_start, homotopySlice_finish] using hi
-  unfold compactIndexPairing
-  rw [he]
-
-end CoarseAssembly.Analytic
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.CompactIndexPairing
+
+namespace OAI
+
+/-! # Homotopy invariance in the unitary argument of the index pairing -/
+
+noncomputable section
+
+namespace CoarseAssembly.Analytic
+
+open scoped CStarAlgebra
+
+variable {A : Type} [CStarAlgebra A]
+
+theorem rawCycleIndex_unitaryHomotopy (c : OddCycle A) (u : Interval → A)
+    (hu : Continuous u) (hul : ∀ t, star (u t) * u t = 1)
+    (hur : ∀ t, u t * star (u t) = 1) :
+    rawCycleIndex c (u start) = rawCycleIndex c (u finish) := by
+  apply fredholmIndex_homotopy (fun t => rawCycleToeplitz c (u t))
+  · have hrep : Continuous (fun t => c.representation (u t)) := (map_continuous c.representation).comp hu
+    have hext : Continuous (fun t => unitaryExtension c (u t)) := hrep.add continuous_const
+    exact ((continuous_const.mul hext).mul continuous_const).add continuous_const
+  · intro t
+    exact rawCycleToeplitz_fredholm c (u t) (hul t) (hur t)
+
+theorem indexPairing_unitaryHomotopy (u : Interval → A) (hu : Continuous u)
+    (hul : ∀ t, star (u t) * u t = 1) (hur : ∀ t, u t * star (u t) = 1) :
+    indexPairing (u start) (hul start) (hur start) =
+      indexPairing (u finish) (hul finish) (hur finish) := by
+  apply hom_ext
+  intro c
+  simp only [indexPairing_cycleClass]
+  exact rawCycleIndex_unitaryHomotopy c u hu hul hur
+
+open scoped ZeroAtInfty
+
+variable {X : Type} [TopologicalSpace X] [CompactSpace X]
+
+theorem compactIndexPairing_homotopy {f g : C(X, Circle)} (F : f.Homotopy g) :
+    compactIndexPairing f = compactIndexPairing g := by
+  let u : Interval → C(X, ℂ) := fun t => compactUnitary (homotopySlice F t)
+  have hu : Continuous u := by
+    let coeCircle : C(Circle, ℂ) := ⟨fun z => (z : ℂ), continuous_subtype_val⟩
+    exact (ContinuousMap.continuous_postcomp coeCircle).comp F.toContinuousMap.curry.continuous
+  have hul (t : Interval) : star (u t) * u t = 1 := compactUnitary_star_mul _
+  have hur (t : Interval) : u t * star (u t) = 1 := compactUnitary_mul_star _
+  have hi := indexPairing_unitaryHomotopy u hu hul hur
+  have he : indexPairing (compactUnitary f) (compactUnitary_star_mul f)
+      (compactUnitary_mul_star f) = indexPairing (compactUnitary g)
+      (compactUnitary_star_mul g) (compactUnitary_mul_star g) := by
+    simpa only [u, homotopySlice_start, homotopySlice_finish] using hi
+  unfold compactIndexPairing
+  rw [he]
+
+end CoarseAssembly.Analytic
+
+end
+
+end OAI

lean/OAI/Topology/CoarseAssembly/UnitaryIndex.lean

@@ -1,76 +1,76 @@
-import OAI.Topology.CoarseAssembly.DegenerateCutDown
-
-namespace OAI
-
-/-! # Unitary invariance of the explicit cycle index -/
-
-noncomputable section
-
-namespace CoarseAssembly.Analytic
-
-variable {H K : HilbertSpace}
-
-theorem operatorConjugation_equiv (e : H ≃ₗᵢ[ℂ] K) (T : End H) :
-    operatorConjugation e.toLinearIsometry T = e.conjStarAlgEquiv T := by
-  have he : e.toLinearIsometry.toContinuousLinearMap.adjoint =
-      (e.symm : K →L[ℂ] H) := e.adjoint_eq_symm
-  ext x
-  change e (T (e.toLinearIsometry.toContinuousLinearMap.adjoint x)) = e (T (e.symm x))
-  rw [he]
-  rfl
-
-theorem symmetricCutDown_map (φ : End H →⋆ₐ[ℂ] End K) (P F : End H) :
-    φ (symmetricCutDown P F) = symmetricCutDown (φ P) (φ F) := by
-  have hs (r : ℝ) (T : End H) : φ (r • T) = r • φ T := φ.toAlgHom.map_smul_of_tower r T
-  unfold symmetricCutDown
-  rw [map_add, hs, map_add, map_star,
-    map_mul, map_mul, map_sub, map_one]
-
-theorem positiveProjection_map (φ : End H →⋆ₐ[ℂ] End K) (F : End H) :
-    φ (positiveProjection F) = positiveProjection (φ F) := by
-  have hs (r : ℝ) (T : End H) : φ (r • T) = r • φ T := φ.toAlgHom.map_smul_of_tower r T
-  unfold positiveProjection
-  rw [hs, map_add, map_one]
-
-theorem toeplitzCompression_map (φ : End H →⋆ₐ[ℂ] End K) (P U : End H) :
-    φ (toeplitzCompression P U) = toeplitzCompression (φ P) (φ U) := by
-  simp only [toeplitzCompression, map_add, map_mul, map_sub, map_one]
-
-theorem fredholmIndex_conjugation (e : H ≃ₗᵢ[ℂ] K) (T : End H) (hT : T.IsFredholm) :
-    (e.conjStarAlgEquiv T).toLinearMap.index = T.toLinearMap.index := by
-  have he := e.toContinuousLinearEquiv.isFredholm
-  have hei := e.symm.toContinuousLinearEquiv.isFredholm
-  have hz : (e.toContinuousLinearEquiv : H →L[ℂ] K).toLinearMap.index = 0 :=
-    LinearMap.index_of_bijective e.bijective
-  have hzi : (e.symm.toContinuousLinearEquiv : K →L[ℂ] H).toLinearMap.index = 0 :=
-    LinearMap.index_of_bijective e.symm.bijective
-  change ((e.toContinuousLinearEquiv : H →L[ℂ] K) ∘L T ∘L
-    (e.symm.toContinuousLinearEquiv : K →L[ℂ] H)).toLinearMap.index = _
-  rw [he.index_comp (hT.comp hei), hT.index_comp hei, hz, hzi, zero_add, add_zero]
-
-variable {A : Type} [CStarAlgebra A]
-
-theorem rawCycleToeplitz_unitary (c d : OddCycle A) (h : c.UnitaryEquivalent d) (u : A) :
-    rawCycleToeplitz d u = h.unitary.conjStarAlgEquiv (rawCycleToeplitz c u) := by
-  let φ : End c.space →⋆ₐ[ℂ] End d.space := h.unitary.conjStarAlgEquiv.toStarAlgHom
-  have hrep (a : A) : d.representation a = φ (c.representation a) :=
-    (h.representation_eq a).trans (operatorConjugation_equiv h.unitary _)
-  have hop : d.operator = φ c.operator :=
-    h.operator_eq.trans (operatorConjugation_equiv h.unitary _)
-  change rawCycleToeplitz d u = φ (rawCycleToeplitz c u)
-  unfold rawCycleToeplitz
-  rw [toeplitzCompression_map φ, positiveProjection_map φ, symmetricCutDown_map φ]
-  simp only [unitaryExtension, map_add, map_sub, map_one]
-  rw [hop, hrep 1, hrep u]
-
-theorem rawCycleIndex_unitary (c d : OddCycle A) (h : c.UnitaryEquivalent d) (u : A)
-    (hul : star u * u = 1) (hur : u * star u = 1) : rawCycleIndex d u = rawCycleIndex c u := by
-  unfold rawCycleIndex
-  rw [rawCycleToeplitz_unitary c d h u]
-  exact fredholmIndex_conjugation h.unitary _ (rawCycleToeplitz_fredholm c u hul hur)
-
-end CoarseAssembly.Analytic
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.DegenerateCutDown
+
+namespace OAI
+
+/-! # Unitary invariance of the explicit cycle index -/
+
+noncomputable section
+
+namespace CoarseAssembly.Analytic
+
+variable {H K : HilbertSpace}
+
+theorem operatorConjugation_equiv (e : H ≃ₗᵢ[ℂ] K) (T : End H) :
+    operatorConjugation e.toLinearIsometry T = e.conjStarAlgEquiv T := by
+  have he : e.toLinearIsometry.toContinuousLinearMap.adjoint =
+      (e.symm : K →L[ℂ] H) := e.adjoint_eq_symm
+  ext x
+  change e (T (e.toLinearIsometry.toContinuousLinearMap.adjoint x)) = e (T (e.symm x))
+  rw [he]
+  rfl
+
+theorem symmetricCutDown_map (φ : End H →⋆ₐ[ℂ] End K) (P F : End H) :
+    φ (symmetricCutDown P F) = symmetricCutDown (φ P) (φ F) := by
+  have hs (r : ℝ) (T : End H) : φ (r • T) = r • φ T := φ.toAlgHom.map_smul_of_tower r T
+  unfold symmetricCutDown
+  rw [map_add, hs, map_add, map_star,
+    map_mul, map_mul, map_sub, map_one]
+
+theorem positiveProjection_map (φ : End H →⋆ₐ[ℂ] End K) (F : End H) :
+    φ (positiveProjection F) = positiveProjection (φ F) := by
+  have hs (r : ℝ) (T : End H) : φ (r • T) = r • φ T := φ.toAlgHom.map_smul_of_tower r T
+  unfold positiveProjection
+  rw [hs, map_add, map_one]
+
+theorem toeplitzCompression_map (φ : End H →⋆ₐ[ℂ] End K) (P U : End H) :
+    φ (toeplitzCompression P U) = toeplitzCompression (φ P) (φ U) := by
+  simp only [toeplitzCompression, map_add, map_mul, map_sub, map_one]
+
+theorem fredholmIndex_conjugation (e : H ≃ₗᵢ[ℂ] K) (T : End H) (hT : T.IsFredholm) :
+    (e.conjStarAlgEquiv T).toLinearMap.index = T.toLinearMap.index := by
+  have he := e.toContinuousLinearEquiv.isFredholm
+  have hei := e.symm.toContinuousLinearEquiv.isFredholm
+  have hz : (e.toContinuousLinearEquiv : H →L[ℂ] K).toLinearMap.index = 0 :=
+    LinearMap.index_of_bijective e.bijective
+  have hzi : (e.symm.toContinuousLinearEquiv : K →L[ℂ] H).toLinearMap.index = 0 :=
+    LinearMap.index_of_bijective e.symm.bijective
+  change ((e.toContinuousLinearEquiv : H →L[ℂ] K) ∘L T ∘L
+    (e.symm.toContinuousLinearEquiv : K →L[ℂ] H)).toLinearMap.index = _
+  rw [he.index_comp (hT.comp hei), hT.index_comp hei, hz, hzi, zero_add, add_zero]
+
+variable {A : Type} [CStarAlgebra A]
+
+theorem rawCycleToeplitz_unitary (c d : OddCycle A) (h : c.UnitaryEquivalent d) (u : A) :
+    rawCycleToeplitz d u = h.unitary.conjStarAlgEquiv (rawCycleToeplitz c u) := by
+  let φ : End c.space →⋆ₐ[ℂ] End d.space := h.unitary.conjStarAlgEquiv.toStarAlgHom
+  have hrep (a : A) : d.representation a = φ (c.representation a) :=
+    (h.representation_eq a).trans (operatorConjugation_equiv h.unitary _)
+  have hop : d.operator = φ c.operator :=
+    h.operator_eq.trans (operatorConjugation_equiv h.unitary _)
+  change rawCycleToeplitz d u = φ (rawCycleToeplitz c u)
+  unfold rawCycleToeplitz
+  rw [toeplitzCompression_map φ, positiveProjection_map φ, symmetricCutDown_map φ]
+  simp only [unitaryExtension, map_add, map_sub, map_one]
+  rw [hop, hrep 1, hrep u]
+
+theorem rawCycleIndex_unitary (c d : OddCycle A) (h : c.UnitaryEquivalent d) (u : A)
+    (hul : star u * u = 1) (hur : u * star u = 1) : rawCycleIndex d u = rawCycleIndex c u := by
+  unfold rawCycleIndex
+  rw [rawCycleToeplitz_unitary c d h u]
+  exact fredholmIndex_conjugation h.unitary _ (rawCycleToeplitz_fredholm c u hul hur)
+
+end CoarseAssembly.Analytic
+
+end
+
+end OAI

lean/OAI/Topology/CoarseAssembly/UnrestrictedBoundedHomotopy.lean

@@ -1,80 +1,80 @@
-import OAI.Topology.CoarseAssembly.BoundedFamilySubdivision
-
-namespace OAI
-
-/-! # Whole-union boundary invariance without uniform homotopy speeds
-
-Finite subdivisions are chosen separately on each component. The two sums
-of consecutive samples have uniformly close localized operators. Rotation
-and direct-sum additivity cancel the common interior samples in actual Roe K₁.
--/
-
-noncomputable section
-
-namespace CoarseAssembly.Rips
-
-open Analytic DiscreteRoe
-
-variable {Q : ℕ → Type} [∀ i, MetricSpace (Q i)] [∀ i, Fintype (Q i)]
-  [∀ i, DecidableEq (Q i)] {r : ℝ}
-
-namespace BoundedFamilyHomotopy
-
-theorem sampled_telescope (h : BoundedFamilyHomotopy Q r)
-    (χ : ∀ i, C(Point (Q i) r, ℂ) →⋆ₐ[ℂ] ℂ)
-    (o : ∀ i, Q i) (hr : 0 ≤ r) (N : ℕ → ℕ) (τ : ℕ → ℕ → Interval)
-    (hnear : (h.sample χ (fun i => N i + 1) τ).boundary o hr =
-      (h.sample χ (fun i => N i + 1) (fun i j => τ i (j + 1))).boundary o hr) :
-    (h.eval (fun i => τ i 0)).boundary o hr =
-      (h.eval (fun i => τ i (N i + 1))).boundary o hr := by
-  let := separatedMetric o
-  have hleft := h.sample_head_tail_boundary χ o hr N τ
-  have hright := h.sample_head_tail_boundary χ o hr N
-    (fun i => lastEndpointFirst (N i) (τ i))
-  have hh : (fun i => lastEndpointFirst (N i) (τ i) 0) =
-      (fun i => τ i (N i + 1)) := funext (fun i => lastEndpointFirst_zero (N i) (τ i))
-  have ht : (fun i j => lastEndpointFirst (N i) (τ i) (j + 1)) =
-      (fun i j => τ i (j + 1)) := funext (fun i => funext (fun j => lastEndpointFirst_succ (N i) (τ i) j))
-  have hh' := congrArg (fun t : ℕ → Interval => (h.eval t).boundary o hr) hh
-  have ht' := congrArg (fun t : ℕ → ℕ → Interval => (h.sample χ N t).boundary o hr) ht
-  have hright' := hright.trans (congrArg₂ (fun a b => a + b) hh' ht')
-  have hrotate := h.sample_rotate_boundary χ o hr N τ
-  exact add_right_cancel (hleft.symm.trans (hnear.trans (hrotate.symm.trans hright')))
-
-theorem endpoint_boundary_eq (h : BoundedFamilyHomotopy Q r)
-    (χ : ∀ i, C(Point (Q i) r, ℂ) →⋆ₐ[ℂ] ℂ)
-    (o : ∀ i, Q i) (hr : 0 ≤ r) :
-    (h.eval (fun _ => start)).boundary o hr = (h.eval (fun _ => finish)).boundary o hr := by
-  let s := h.smallSubdivision
-  have he := h.sampled_telescope χ o hr s.count s.time
-    (h.sampled_boundary_eq χ o hr s.count s.time s.step)
-  have h0 : (fun i => s.time i 0) = (fun _ => start) := funext s.at_start
-  have h1 : (fun i => s.time i (s.count i + 1)) = (fun _ => finish) := funext s.at_finish
-  have he0 := congrArg (fun t : ℕ → Interval => (h.eval t).boundary o hr) h0
-  have he1 := congrArg (fun t : ℕ → Interval => (h.eval t).boundary o hr) h1
-  exact he0.symm.trans (he.trans he1)
-
-end BoundedFamilyHomotopy
-
-/-- Arbitrary componentwise continuous unital self-adjoint homotopies with
-a common operator bound preserve the whole-union Roe boundary. -/
-theorem ampleRipsFamilyBoundary_boundedHomotopy
-    (o : ∀ i, Q i) (hr : 0 ≤ r)
-    (χ : ∀ i, C(Point (Q i) r, ℂ) →⋆ₐ[ℂ] ℂ)
-    (h : ∀ i, OddHomotopy C(Point (Q i) r, ℂ))
-    (hπ : ∀ t i, (h i).representation t 1 = 1)
-    (hs : ∀ t i, IsSelfAdjoint ((h i).operator t))
-    (C : ℝ) (hC : 1 ≤ C) (hn : ∀ t i, ‖(h i).operator t‖ ≤ C) :
-    letI := separatedMetric o
-    ampleRipsFamilyBoundary o r hr (fun i => (h i).eval start) (hπ start) (hs start)
-      C (zero_le_one.trans hC) (hn start) =
-    ampleRipsFamilyBoundary o r hr (fun i => (h i).eval finish) (hπ finish) (hs finish)
-      C (zero_le_one.trans hC) (hn finish) := by
-  let H : BoundedFamilyHomotopy Q r := ⟨h, hπ, hs, C, hC, hn⟩
-  exact H.endpoint_boundary_eq χ o hr
-
-end CoarseAssembly.Rips
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.BoundedFamilySubdivision
+
+namespace OAI
+
+/-! # Whole-union boundary invariance without uniform homotopy speeds
+
+Finite subdivisions are chosen separately on each component. The two sums
+of consecutive samples have uniformly close localized operators. Rotation
+and direct-sum additivity cancel the common interior samples in actual Roe K₁.
+-/
+
+noncomputable section
+
+namespace CoarseAssembly.Rips
+
+open Analytic DiscreteRoe
+
+variable {Q : ℕ → Type} [∀ i, MetricSpace (Q i)] [∀ i, Fintype (Q i)]
+  [∀ i, DecidableEq (Q i)] {r : ℝ}
+
+namespace BoundedFamilyHomotopy
+
+theorem sampled_telescope (h : BoundedFamilyHomotopy Q r)
+    (χ : ∀ i, C(Point (Q i) r, ℂ) →⋆ₐ[ℂ] ℂ)
+    (o : ∀ i, Q i) (hr : 0 ≤ r) (N : ℕ → ℕ) (τ : ℕ → ℕ → Interval)
+    (hnear : (h.sample χ (fun i => N i + 1) τ).boundary o hr =
+      (h.sample χ (fun i => N i + 1) (fun i j => τ i (j + 1))).boundary o hr) :
+    (h.eval (fun i => τ i 0)).boundary o hr =
+      (h.eval (fun i => τ i (N i + 1))).boundary o hr := by
+  let := separatedMetric o
+  have hleft := h.sample_head_tail_boundary χ o hr N τ
+  have hright := h.sample_head_tail_boundary χ o hr N
+    (fun i => lastEndpointFirst (N i) (τ i))
+  have hh : (fun i => lastEndpointFirst (N i) (τ i) 0) =
+      (fun i => τ i (N i + 1)) := funext (fun i => lastEndpointFirst_zero (N i) (τ i))
+  have ht : (fun i j => lastEndpointFirst (N i) (τ i) (j + 1)) =
+      (fun i j => τ i (j + 1)) := funext (fun i => funext (fun j => lastEndpointFirst_succ (N i) (τ i) j))
+  have hh' := congrArg (fun t : ℕ → Interval => (h.eval t).boundary o hr) hh
+  have ht' := congrArg (fun t : ℕ → ℕ → Interval => (h.sample χ N t).boundary o hr) ht
+  have hright' := hright.trans (congrArg₂ (fun a b => a + b) hh' ht')
+  have hrotate := h.sample_rotate_boundary χ o hr N τ
+  exact add_right_cancel (hleft.symm.trans (hnear.trans (hrotate.symm.trans hright')))
+
+theorem endpoint_boundary_eq (h : BoundedFamilyHomotopy Q r)
+    (χ : ∀ i, C(Point (Q i) r, ℂ) →⋆ₐ[ℂ] ℂ)
+    (o : ∀ i, Q i) (hr : 0 ≤ r) :
+    (h.eval (fun _ => start)).boundary o hr = (h.eval (fun _ => finish)).boundary o hr := by
+  let s := h.smallSubdivision
+  have he := h.sampled_telescope χ o hr s.count s.time
+    (h.sampled_boundary_eq χ o hr s.count s.time s.step)
+  have h0 : (fun i => s.time i 0) = (fun _ => start) := funext s.at_start
+  have h1 : (fun i => s.time i (s.count i + 1)) = (fun _ => finish) := funext s.at_finish
+  have he0 := congrArg (fun t : ℕ → Interval => (h.eval t).boundary o hr) h0
+  have he1 := congrArg (fun t : ℕ → Interval => (h.eval t).boundary o hr) h1
+  exact he0.symm.trans (he.trans he1)
+
+end BoundedFamilyHomotopy
+
+/-- Arbitrary componentwise continuous unital self-adjoint homotopies with
+a common operator bound preserve the whole-union Roe boundary. -/
+theorem ampleRipsFamilyBoundary_boundedHomotopy
+    (o : ∀ i, Q i) (hr : 0 ≤ r)
+    (χ : ∀ i, C(Point (Q i) r, ℂ) →⋆ₐ[ℂ] ℂ)
+    (h : ∀ i, OddHomotopy C(Point (Q i) r, ℂ))
+    (hπ : ∀ t i, (h i).representation t 1 = 1)
+    (hs : ∀ t i, IsSelfAdjoint ((h i).operator t))
+    (C : ℝ) (hC : 1 ≤ C) (hn : ∀ t i, ‖(h i).operator t‖ ≤ C) :
+    letI := separatedMetric o
+    ampleRipsFamilyBoundary o r hr (fun i => (h i).eval start) (hπ start) (hs start)
+      C (zero_le_one.trans hC) (hn start) =
+    ampleRipsFamilyBoundary o r hr (fun i => (h i).eval finish) (hπ finish) (hs finish)
+      C (zero_le_one.trans hC) (hn finish) := by
+  let H : BoundedFamilyHomotopy Q r := ⟨h, hπ, hs, C, hC, hn⟩
+  exact H.endpoint_boundary_eq χ o hr
+
+end CoarseAssembly.Rips
+
+end
+
+end OAI

lean/OAI/Topology/CoarseAssembly/WindingHomotopy.lean

@@ -1,83 +1,83 @@
-import OAI.Topology.CoarseAssembly.MultiplicativeIndexPairing
-
-namespace OAI
-
-/-! # The explicit homotopy from a lifted loop to its winding power -/
-
-noncomputable section
-
-namespace CoarseAssembly
-
-open unitInterval
-
-/-- The degree-`n` self-map of the additive circle. -/
-def circleDegree (n : ℤ) : C(WindingCircle, WindingCircle) :=
-  ⟨fun z => n • z, continuous_id.zsmul n⟩
-
-theorem windingCircle_intCast (n : ℤ) : ((n : ℝ) : WindingCircle) = 0 := by
-  have h : ((n • (1 : ℝ) : ℝ) : WindingCircle) = 0 := by
-    rw [AddCircle.coe_zsmul, AddCircle.coe_period, smul_zero]
-  simpa only [zsmul_eq_mul, mul_one] using h
-
-/-- Straight-line interpolation of the real lift with `t ↦ n*t`. -/
-def windingInterpolation (n : ℤ) (L : Path (0 : ℝ) (n : ℝ)) (s : unitInterval) :
-    Path (0 : WindingCircle) 0 where
-  toFun t := (((1 - (s : ℝ)) * L t + (s : ℝ) * n * (t : ℝ) : ℝ) : WindingCircle)
-  continuous_toFun := (AddCircle.continuous_mk' 1).comp
-    ((L.continuous.const_mul _).add (continuous_subtype_val.const_mul _))
-  source' := by simp
-  target' := by
-    have he : (1 - (s : ℝ)) * (n : ℝ) + (s : ℝ) * n * 1 = n := by ring
-    simp only [Path.target, Set.Icc.coe_one, he]
-    exact windingCircle_intCast n
-
-def windingInterpolationMap (n : ℤ) (L : Path (0 : ℝ) (n : ℝ)) :
-    unitInterval × WindingCircle → WindingCircle :=
-  fun p => loopCircle (windingInterpolation n L p.1) p.2
-
-theorem windingInterpolationMap_continuous (n : ℤ) (L : Path (0 : ℝ) (n : ℝ)) :
-    Continuous (windingInterpolationMap n L) := by
-  let q : unitInterval × unitInterval → unitInterval × WindingCircle :=
-    fun p => (p.1, ((p.2 : ℝ) : WindingCircle))
-  have hq : Continuous q := continuous_fst.prodMk
-    ((AddCircle.continuous_mk' 1).comp (continuous_subtype_val.comp continuous_snd))
-  have hqs : Function.Surjective q := by
-    rintro ⟨s, z⟩
-    obtain ⟨t, ht⟩ := circle_parameter_surjective z
-    exact ⟨(s, t), Prod.ext rfl ht⟩
-  apply ((hq.isClosedMap.isQuotientMap hq hqs).continuous_iff).2
-  have he : windingInterpolationMap n L ∘ q = fun p : unitInterval × unitInterval =>
-      (((1 - (p.1 : ℝ)) * L p.2 + (p.1 : ℝ) * n * (p.2 : ℝ) : ℝ) : WindingCircle) := by
-    funext p
-    exact loopCircle_apply (windingInterpolation n L p.1) p.2
-  rw [he]
-  apply (AddCircle.continuous_mk' 1).comp
-  exact ((continuous_const.sub (continuous_subtype_val.comp continuous_fst)).mul
-    (L.continuous.comp continuous_snd)).add
-      (((continuous_subtype_val.comp continuous_fst).mul continuous_const).mul
-        (continuous_subtype_val.comp continuous_snd))
-
-def windingHomotopy (g : C(WindingCircle, WindingCircle)) (n : ℤ)
-    (L : Path (0 : ℝ) (n : ℝ))
-    (hL : ∀ t : unitInterval, g (t : ℝ) = (L t : WindingCircle)) :
-    g.Homotopy (circleDegree n) where
-  toFun := windingInterpolationMap n L
-  continuous_toFun := windingInterpolationMap_continuous n L
-  map_zero_left z := by
-    obtain ⟨t, rfl⟩ := circle_parameter_surjective z
-    rw [windingInterpolationMap, loopCircle_apply, hL t]
-    change ((((1 - (0 : ℝ)) * L t + (0 : ℝ) * n * (t : ℝ)) : ℝ) : WindingCircle) = _
-    simp
-  map_one_left z := by
-    obtain ⟨t, rfl⟩ := circle_parameter_surjective z
-    rw [windingInterpolationMap, loopCircle_apply]
-    change ((((1 - (1 : ℝ)) * L t + (1 : ℝ) * n * (t : ℝ)) : ℝ) : WindingCircle) =
-      n • ((t : ℝ) : WindingCircle)
-    simpa only [sub_self, zero_mul, zero_add, one_mul, zsmul_eq_mul] using
-      (AddCircle.coe_zsmul (p := (1 : ℝ)) (n := n) (x := (t : ℝ)))
-
-end CoarseAssembly
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.MultiplicativeIndexPairing
+
+namespace OAI
+
+/-! # The explicit homotopy from a lifted loop to its winding power -/
+
+noncomputable section
+
+namespace CoarseAssembly
+
+open unitInterval
+
+/-- The degree-`n` self-map of the additive circle. -/
+def circleDegree (n : ℤ) : C(WindingCircle, WindingCircle) :=
+  ⟨fun z => n • z, continuous_id.zsmul n⟩
+
+theorem windingCircle_intCast (n : ℤ) : ((n : ℝ) : WindingCircle) = 0 := by
+  have h : ((n • (1 : ℝ) : ℝ) : WindingCircle) = 0 := by
+    rw [AddCircle.coe_zsmul, AddCircle.coe_period, smul_zero]
+  simpa only [zsmul_eq_mul, mul_one] using h
+
+/-- Straight-line interpolation of the real lift with `t ↦ n*t`. -/
+def windingInterpolation (n : ℤ) (L : Path (0 : ℝ) (n : ℝ)) (s : unitInterval) :
+    Path (0 : WindingCircle) 0 where
+  toFun t := (((1 - (s : ℝ)) * L t + (s : ℝ) * n * (t : ℝ) : ℝ) : WindingCircle)
+  continuous_toFun := (AddCircle.continuous_mk' 1).comp
+    ((L.continuous.const_mul _).add (continuous_subtype_val.const_mul _))
+  source' := by simp
+  target' := by
+    have he : (1 - (s : ℝ)) * (n : ℝ) + (s : ℝ) * n * 1 = n := by ring
+    simp only [Path.target, Set.Icc.coe_one, he]
+    exact windingCircle_intCast n
+
+def windingInterpolationMap (n : ℤ) (L : Path (0 : ℝ) (n : ℝ)) :
+    unitInterval × WindingCircle → WindingCircle :=
+  fun p => loopCircle (windingInterpolation n L p.1) p.2
+
+theorem windingInterpolationMap_continuous (n : ℤ) (L : Path (0 : ℝ) (n : ℝ)) :
+    Continuous (windingInterpolationMap n L) := by
+  let q : unitInterval × unitInterval → unitInterval × WindingCircle :=
+    fun p => (p.1, ((p.2 : ℝ) : WindingCircle))
+  have hq : Continuous q := continuous_fst.prodMk
+    ((AddCircle.continuous_mk' 1).comp (continuous_subtype_val.comp continuous_snd))
+  have hqs : Function.Surjective q := by
+    rintro ⟨s, z⟩
+    obtain ⟨t, ht⟩ := circle_parameter_surjective z
+    exact ⟨(s, t), Prod.ext rfl ht⟩
+  apply ((hq.isClosedMap.isQuotientMap hq hqs).continuous_iff).2
+  have he : windingInterpolationMap n L ∘ q = fun p : unitInterval × unitInterval =>
+      (((1 - (p.1 : ℝ)) * L p.2 + (p.1 : ℝ) * n * (p.2 : ℝ) : ℝ) : WindingCircle) := by
+    funext p
+    exact loopCircle_apply (windingInterpolation n L p.1) p.2
+  rw [he]
+  apply (AddCircle.continuous_mk' 1).comp
+  exact ((continuous_const.sub (continuous_subtype_val.comp continuous_fst)).mul
+    (L.continuous.comp continuous_snd)).add
+      (((continuous_subtype_val.comp continuous_fst).mul continuous_const).mul
+        (continuous_subtype_val.comp continuous_snd))
+
+def windingHomotopy (g : C(WindingCircle, WindingCircle)) (n : ℤ)
+    (L : Path (0 : ℝ) (n : ℝ))
+    (hL : ∀ t : unitInterval, g (t : ℝ) = (L t : WindingCircle)) :
+    g.Homotopy (circleDegree n) where
+  toFun := windingInterpolationMap n L
+  continuous_toFun := windingInterpolationMap_continuous n L
+  map_zero_left z := by
+    obtain ⟨t, rfl⟩ := circle_parameter_surjective z
+    rw [windingInterpolationMap, loopCircle_apply, hL t]
+    change ((((1 - (0 : ℝ)) * L t + (0 : ℝ) * n * (t : ℝ)) : ℝ) : WindingCircle) = _
+    simp
+  map_one_left z := by
+    obtain ⟨t, rfl⟩ := circle_parameter_surjective z
+    rw [windingInterpolationMap, loopCircle_apply]
+    change ((((1 - (1 : ℝ)) * L t + (1 : ℝ) * n * (t : ℝ)) : ℝ) : WindingCircle) =
+      n • ((t : ℝ) : WindingCircle)
+    simpa only [sub_self, zero_mul, zero_add, one_mul, zsmul_eq_mul] using
+      (AddCircle.coe_zsmul (p := (1 : ℝ)) (n := n) (x := (t : ℝ)))
+
+end CoarseAssembly
+
+end
+
+end OAI

lean/OAI/Topology/CoarseAssembly/WindingIndexEvaluation.lean

@@ -1,60 +1,60 @@
-import OAI.Topology.CoarseAssembly.WindingHomotopy
-
-namespace OAI
-
-/-! # Reducing the winding index formula to one generator evaluation
-
-The pairing, its naturality, homotopy invariance and power law are proved.
-Only an actual circle class with coordinate index one remains to construct.
--/
-
-noncomputable section
-
-namespace CoarseAssembly
-
-open Analytic unitInterval
-
-theorem circleUnitary_degree (n : ℤ) : circleUnitary (circleDegree n) =
-    circleUnitary (ContinuousMap.id WindingCircle) ^ n := by
-  apply ContinuousMap.ext
-  intro z
-  exact AddCircle.toCircle_zsmul z n
-
-theorem winding_index_hom (g : C(WindingCircle, WindingCircle)) (n : ℤ)
-    (L : Path (0 : ℝ) (n : ℝ))
-    (hL : ∀ t : unitInterval, g (t : ℝ) = (L t : WindingCircle)) :
-    compactIndexPairing (circleUnitary g) =
-      n • compactIndexPairing (circleUnitary (ContinuousMap.id WindingCircle)) := by
-  let toCircleMap : C(WindingCircle, Circle) := ⟨AddCircle.toCircle, AddCircle.continuous_toCircle⟩
-  have h := compactIndexPairing_homotopy
-    ((ContinuousMap.Homotopy.refl toCircleMap).comp (windingHomotopy g n L hL))
-  change compactIndexPairing (circleUnitary g) = compactIndexPairing (circleUnitary (circleDegree n)) at h
-  rw [circleUnitary_degree, compactIndexPairing_zpow] at h
-  exact h
-
-theorem winding_index (g : C(WindingCircle, WindingCircle)) (n : ℤ)
-    (L : Path (0 : ℝ) (n : ℝ))
-    (hL : ∀ t : unitInterval, g (t : ℝ) = (L t : WindingCircle))
-    (α : LocallyFiniteK1 WindingCircle) :
-    compactIndexPairing (circleUnitary g) α =
-      n * compactIndexPairing (circleUnitary (ContinuousMap.id WindingCircle)) α := by
-  simpa only [AddMonoidHom.smul_apply, zsmul_eq_mul, Int.cast_id] using
-    DFunLike.congr_fun (winding_index_hom g n L hL) α
-
-/-- All the published circle fields follow from a single coordinate-index-one class. -/
-def circleInputOfIndexOne (α : LocallyFiniteK1 WindingCircle)
-    (hα : compactIndexPairing (circleUnitary (ContinuousMap.id WindingCircle)) α = 1) :
-    PublishedCircleInput where
-  fundamental := α
-  pairing f := compactIndexPairing (circleUnitary f)
-  loop_naturality f γ := by
-    have h := DFunLike.congr_fun (compactIndexPairing_natural (circleUnitary f) γ) α
-    exact h
-  winding g n L hL := by
-    rw [winding_index g n L hL α, hα, mul_one]
-
-end CoarseAssembly
-
-end
-
-end OAI
+import OAI.Topology.CoarseAssembly.WindingHomotopy
+
+namespace OAI
+
+/-! # Reducing the winding index formula to one generator evaluation
+
+The pairing, its naturality, homotopy invariance and power law are proved.
+Only an actual circle class with coordinate index one remains to construct.
+-/
+
+noncomputable section
+
+namespace CoarseAssembly
+
+open Analytic unitInterval
+
+theorem circleUnitary_degree (n : ℤ) : circleUnitary (circleDegree n) =
+    circleUnitary (ContinuousMap.id WindingCircle) ^ n := by
+  apply ContinuousMap.ext
+  intro z
+  exact AddCircle.toCircle_zsmul z n
+
+theorem winding_index_hom (g : C(WindingCircle, WindingCircle)) (n : ℤ)
+    (L : Path (0 : ℝ) (n : ℝ))
+    (hL : ∀ t : unitInterval, g (t : ℝ) = (L t : WindingCircle)) :
+    compactIndexPairing (circleUnitary g) =
+      n • compactIndexPairing (circleUnitary (ContinuousMap.id WindingCircle)) := by
+  let toCircleMap : C(WindingCircle, Circle) := ⟨AddCircle.toCircle, AddCircle.continuous_toCircle⟩
+  have h := compactIndexPairing_homotopy
+    ((ContinuousMap.Homotopy.refl toCircleMap).comp (windingHomotopy g n L hL))
+  change compactIndexPairing (circleUnitary g) = compactIndexPairing (circleUnitary (circleDegree n)) at h
+  rw [circleUnitary_degree, compactIndexPairing_zpow] at h
+  exact h
+
+theorem winding_index (g : C(WindingCircle, WindingCircle)) (n : ℤ)
+    (L : Path (0 : ℝ) (n : ℝ))
+    (hL : ∀ t : unitInterval, g (t : ℝ) = (L t : WindingCircle))
+    (α : LocallyFiniteK1 WindingCircle) :
+    compactIndexPairing (circleUnitary g) α =
+      n * compactIndexPairing (circleUnitary (ContinuousMap.id WindingCircle)) α := by
+  simpa only [AddMonoidHom.smul_apply, zsmul_eq_mul, Int.cast_id] using
+    DFunLike.congr_fun (winding_index_hom g n L hL) α
+
+/-- All the published circle fields follow from a single coordinate-index-one class. -/
+def circleInputOfIndexOne (α : LocallyFiniteK1 WindingCircle)
+    (hα : compactIndexPairing (circleUnitary (ContinuousMap.id WindingCircle)) α = 1) :
+    PublishedCircleInput where
+  fundamental := α
+  pairing f := compactIndexPairing (circleUnitary f)
+  loop_naturality f γ := by
+    have h := DFunLike.congr_fun (compactIndexPairing_natural (circleUnitary f) γ) α
+    exact h
+  winding g n L hL := by
+    rw [winding_index g n L hL α, hα, mul_one]
+
+end CoarseAssembly
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/AffineTransport.lean

@@ -1,449 +1,449 @@
-import OAI.Topology.EilenbergGanea.SmallHolonomy
-import OAI.GroupTheory.RightAngledArtin.TraceWords
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set Filter Topology MeasureTheory
-open scoped Quaternion ContDiff
-
-namespace EilenbergGanea
-namespace AffineTransport
-
-/-- A finite family of proper quaternion subspaces cannot cover ℍ. -/
-theorem exists_avoid_subspaces {I : Type*} [Finite I]
-    (p : I → Submodule ℝ ℍ) (hp : ∀ i, p i ≠ ⊤) :
-    ∃ q : ℍ, ∀ i, q ∉ p i := by
-  have ho (i) : IsOpen ((p i : Set ℍ)ᶜ) :=
-    (p i).closed_of_finiteDimensional.isOpen_compl
-  have hd (i) : Dense ((p i : Set ℍ)ᶜ) := by
-    apply interior_eq_empty_iff_dense_compl.mp
-    apply Set.not_nonempty_iff_eq_empty.mp
-    intro h
-    exact hp i ((p i).eq_top_of_nonempty_interior' h)
-  obtain ⟨q,hq⟩ := (dense_iInter_of_isOpen ho hd).nonempty
-  exact ⟨q,Set.mem_iInter.mp hq⟩
-
-/-- Simultaneous general position for finitely many local quaternion gauges.
-Different vertices may use different invertible linear transformations in each
-chart. No flatness or topology is assumed in this finite-dimensional lemma. -/
-theorem exists_triangle_general_position {A V : Type*} [Finite A] [Fintype V]
-    (T : A → V → ℍ ≃ₗ[ℝ] ℍ) :
-    ∃ q : V → ℍ, ∀ a (s : Finset V), s.card ≤ 3 →
-      LinearIndepOn ℝ (fun v => T a v (q v)) (s : Set V) := by
-  classical
-  suffices ∀ U : Finset V, ∃ q : V → ℍ, ∀ a (s : Finset V), s ⊆ U → s.card ≤ 3 →
-      LinearIndepOn ℝ (fun v => T a v (q v)) (s : Set V) by
-    obtain ⟨q,hq⟩ := this Finset.univ
-    exact ⟨q,fun a s h => hq a s (Finset.subset_univ _) h⟩
-  intro U
-  induction U using Finset.induction with
-  | empty =>
-      refine ⟨0,?_⟩
-      intro a s hs _
-      have : s = ∅ := Finset.subset_empty.mp hs
-      subst s
-      simpa only [Finset.coe_empty,Pi.zero_apply] using
-        linearIndepOn_empty ℝ (fun v => T a v (0 : ℍ))
-  | @insert v U hv ih =>
-      obtain ⟨q,hq⟩ := ih
-      let I := A × {s : Finset V // s ⊆ U ∧ s.card ≤ 2}
-      let p : I → Submodule ℝ ℍ := fun i =>
-        (Submodule.span ℝ ((fun w => T i.1 w (q w)) '' (i.2.val : Set V))).comap
-          (T i.1 v).toLinearMap
-      have hp (i : I) : p i ≠ ⊤ := by
-        have hf : Module.finrank ℝ (Submodule.span ℝ
-            ((fun w => T i.1 w (q w)) '' (i.2.val : Set V))) ≤ 2 := by
-          have hh := finrank_span_finset_le_card (R := ℝ) (i.2.val.image (fun w => T i.1 w (q w)))
-          rw [Finset.coe_image] at hh
-          exact hh.trans ((Finset.card_image_le).trans i.2.property.2)
-        have hproper : Submodule.span ℝ ((fun w => T i.1 w (q w)) ''
-            (i.2.val : Set V)) ≠ ⊤ := by
-          apply ne_of_lt (Submodule.lt_top_of_finrank_lt_finrank _)
-          rw [Quaternion.finrank_eq_four]
-          omega
-        intro hh
-        apply hproper
-        apply top_unique
-        intro x _
-        have hm : (T i.1 v).symm x ∈ p i := by rw [hh]; trivial
-        simpa only [p,Submodule.mem_comap,LinearEquiv.coe_coe,
-          LinearEquiv.apply_symm_apply] using hm
-      obtain ⟨z,hz⟩ := exists_avoid_subspaces p hp
-      refine ⟨Function.update q v z,?_⟩
-      intro a s hs hc
-      by_cases hvs : v ∈ s
-      · have hsub : s.erase v ⊆ U := by
-          intro w hw
-          have hh := hs (Finset.mem_of_mem_erase hw)
-          exact (Finset.mem_insert.mp hh).resolve_left (Finset.ne_of_mem_erase hw)
-        have hcard : (s.erase v).card ≤ 2 := by
-          have he := Finset.card_erase_add_one hvs
-          omega
-        have heq : ∀ w ∈ (s.erase v : Finset V), Function.update q v z w = q w := by
-          intro w hw
-          exact Function.update_of_ne (Finset.ne_of_mem_erase hw) _ _
-        have hi : LinearIndepOn ℝ (fun w => T a w (Function.update q v z w))
-            ((s.erase v : Finset V) : Set V) := by
-          apply (hq a (s.erase v) hsub (by omega)).congr
-          intro w hw
-          dsimp only
-          rw [heq w hw]
-        have hn : T a v (Function.update q v z v) ∉ Submodule.span ℝ
-            ((fun w => T a w (Function.update q v z w)) ''
-              ((s.erase v : Finset V) : Set V)) := by
-          rw [Function.update_self]
-          have him : ((fun w => T a w (Function.update q v z w)) ''
-              ((s.erase v : Finset V) : Set V)) =
-              ((fun w => T a w (q w)) '' ((s.erase v : Finset V) : Set V)) := by
-            apply Set.image_congr
-            intro w hw
-            rw [heq w hw]
-          rw [him]
-          exact hz ⟨a,⟨s.erase v,hsub,hcard⟩⟩
-        have ht := (linearIndepOn_insert (by simp : v ∉ ((s.erase v : Finset V) : Set V))).mpr
-          ⟨hi,hn⟩
-        simpa only [← Finset.coe_insert,Finset.insert_erase hvs] using ht
-      · have hsub : s ⊆ U := by
-          intro w hw
-          have hh := hs hw
-          rcases Finset.mem_insert.mp hh with rfl | hh
-          · exact False.elim (hvs hw)
-          · exact hh
-        apply (hq a s hsub hc).congr
-        intro w hw
-        dsimp only
-        rw [Function.update_of_ne (by intro h; subst w; exact hvs hw)]
-
-
-@[simp] theorem unit_norm (h : SU2) : ‖(h : ℍ)‖ = 1 :=
-  (quaternion_mem_unitary_iff _).mp h.property
-
-def leftLinearEquiv (h : SU2) : ℍ ≃ₗ[ℝ] ℍ where
-  toFun q := (h : ℍ) * q
-  invFun q := (h⁻¹ : SU2) * q
-  left_inv q := by simp [← mul_assoc,← Submonoid.coe_mul]
-  right_inv q := by simp [← mul_assoc,← Submonoid.coe_mul]
-  map_add' _ _ := mul_add _ _ _
-  map_smul' r q := mul_smul_comm r (h : ℍ) q
-
-@[simp] theorem leftLinearEquiv_apply (h : SU2) (q : ℍ) :
-    leftLinearEquiv h q = (h : ℍ) * q := rfl
-
-theorem normalize_left (h : SU2) {q : ℍ} (hq : q ≠ 0) :
-    normalizeQuaternion ((h : ℍ) * q) = h * normalizeQuaternion q := by
-  have hh : (h : ℍ) * q ≠ 0 := by
-    exact (leftLinearEquiv h).map_ne_zero_iff.mpr hq
-  apply Subtype.ext
-  rw [normalizeQuaternion_coe_of_ne_zero hh]
-  change ‖(h : ℍ) * q‖⁻¹ • ((h : ℍ) * q) = (h : ℍ) * (normalizeQuaternion q : ℍ)
-  rw [normalizeQuaternion_coe_of_ne_zero hq,norm_mul,unit_norm,one_mul,mul_smul_comm]
-
-section FiniteSimplex
-variable {V : Type*} [Fintype V]
-
-def coordinateSimplex : Set (V → ℝ) :=
-  {weights | (∀ vertex, 0 ≤ weights vertex) ∧ ∑ vertex, weights vertex = 1}
-
-/-- Closed coordinate face of the standard simplex. -/
-def face (s : Finset V) : Set (V → ℝ) :=
-  {b | b ∈ coordinateSimplex ∧ ∀ v ∉ s, b v = 0}
-
-/-- The whole two-skeleton of a finite standard simplex. -/
-def triangleSet : Set (V → ℝ) :=
-  ⋃ s : {s : Finset V // s.card ≤ 3}, face s.val
-
-abbrev TrianglePoint (V : Type*) [Fintype V] := (triangleSet : Set (V → ℝ))
-
-theorem mem_triangleSet {b : V → ℝ} : b ∈ triangleSet ↔
-    ∃ s : Finset V, s.card ≤ 3 ∧ b ∈ coordinateSimplex ∧ ∀ v ∉ s, b v = 0 := by
-  simp only [triangleSet,Set.mem_iUnion,face,Set.mem_ofPred_eq,Subtype.exists,exists_prop]
-
-omit [Fintype V] in
-private theorem isClosed_face_support (s : Finset V) :
-    IsClosed {b : V → ℝ | ∀ v ∉ s, b v = 0} := by
-  have he : {b : V → ℝ | ∀ v ∉ s, b v = 0} =
-      ⋂ v, ⋂ (_ : v ∉ s), {b : V → ℝ | b v = 0} := by ext b; simp
-  rw [he]
-  exact isClosed_iInter fun v => isClosed_iInter fun _ =>
-    isClosed_eq (continuous_apply v) continuous_const
-
-theorem isCompact_face (s : Finset V) : IsCompact (face s) := by
-  have compact_simplex : IsCompact (coordinateSimplex : Set (V → ℝ)) := by
-    convert isCompact_range
-      (Convexity.StdSimplex.isEmbedding_toFun_comp_weights ℝ V).continuous using 1
-    rw [Convexity.StdSimplex.range_toFun_comp_weights]
-    ext weights
-    simp [coordinateSimplex]
-  exact compact_simplex.inter_right (isClosed_face_support s)
-
-theorem isCompact_triangleSet : IsCompact (triangleSet : Set (V → ℝ)) :=
-  isCompact_iUnion fun _ => isCompact_face _
-
-instance : CompactSpace (TrianglePoint V) :=
-  isCompact_iff_compactSpace.mp isCompact_triangleSet
-
-theorem point_simplex (b : TrianglePoint V) : (b : V → ℝ) ∈ coordinateSimplex := by
-  obtain ⟨_,_,h,_⟩ := mem_triangleSet.mp b.property
-  exact h
-
-theorem point_exists_nonzero (b : TrianglePoint V) : ∃ v, (b : V → ℝ) v ≠ 0 := by
-  by_contra! h
-  have he := (point_simplex b).2
-  simp only [h,Finset.sum_const_zero] at he
-  norm_num at he
-
-variable (h : V → V → SU2) (q : V → ℍ)
-
-def rawGauge (a : V) (b : V → ℝ) : ℍ := ∑ v, b v • ((h a v : ℍ) * q v)
-
-def GeneralPosition : Prop := ∀ a (s : Finset V), s.card ≤ 3 →
-  LinearIndepOn ℝ (fun v => (h a v : ℍ) * q v) (s : Set V)
-
-theorem exists_generalPosition : ∃ q : V → ℍ, GeneralPosition h q := by
-  obtain ⟨q,hq⟩ := exists_triangle_general_position (fun a v => leftLinearEquiv (h a v))
-  exact ⟨q,hq⟩
-
-theorem rawGauge_ne_zero (hq : GeneralPosition h q) (a : V) (b : TrianglePoint V) :
-    rawGauge h q a b ≠ 0 := by
-  classical
-  obtain ⟨s,hs,hb,hout⟩ := mem_triangleSet.mp b.property
-  intro he
-  have he' : ∑ v ∈ s, (b : V → ℝ) v • ((h a v : ℍ) * q v) = 0 := by
-    rw [← he]
-    apply Finset.sum_subset (Finset.subset_univ _)
-    intro v _ hv
-    rw [hout v hv,zero_smul]
-  have hz := (linearIndepOn_iff'.mp (hq a s hs)) s b (Set.Subset.refl _) he'
-  have hall : ∀ v, (b : V → ℝ) v = 0 := by
-    intro v
-    by_cases hv : v ∈ s
-    · exact hz v hv
-    · exact hout v hv
-  have heq := hb.2
-  simp only [hall,Finset.sum_const_zero] at heq
-  norm_num at heq
-
-def gauge (a : V) (b : TrianglePoint V) : SU2 := normalizeQuaternion (rawGauge h q a b)
-
-theorem continuous_rawGauge (a : V) : Continuous (rawGauge h q a) :=
-  continuous_finsetSum _ fun v _ => (continuous_apply v).smul continuous_const
-
-theorem continuous_gauge (hq : GeneralPosition h q) (a : V) : Continuous (gauge h q a) := by
-  apply continuous_iff_continuousAt.mpr
-  intro b
-  exact ContinuousAt.comp (f := fun c : TrianglePoint V => rawGauge h q a c)
-    (normalizeQuaternion_continuousAt (rawGauge_ne_zero h q hq a b))
-    ((continuous_rawGauge h q a).comp continuous_subtype_val).continuousAt
-
-/-- Uniform modulus over all charts, including across the edges of all simplices. -/
-theorem gauges_uniform (hq : GeneralPosition h q) {ε : ℝ} (hε : 0 < ε) :
-    ∃ δ > 0, ∀ b c : TrianglePoint V, dist b c < δ →
-      ∀ a, dist (gauge h q a b) (gauge h q a c) < ε := by
-  have hc : Continuous (fun b : TrianglePoint V => fun a => gauge h q a b) :=
-    continuous_pi fun a => continuous_gauge h q hq a
-  have hu := CompactSpace.uniformContinuous_of_continuous hc
-  obtain ⟨δ,hδ,H⟩ := Metric.uniformContinuous_iff.mp hu ε hε
-  refine ⟨δ,hδ,?_⟩
-  intro b c hbc a
-  exact lt_of_le_of_lt (dist_le_pi_dist (fun a => gauge h q a b) (fun a => gauge h q a c) a) (H hbc)
-
-end FiniteSimplex
-
-theorem unitary_dist_left (a b c : SU2) : dist (a * b) (a * c) = dist b c := by
-  change dist ((a : ℍ) * (b : ℍ)) ((a : ℍ) * (c : ℍ)) = dist (b : ℍ) (c : ℍ)
-  rw [dist_eq_norm,dist_eq_norm,← mul_sub,norm_mul,unit_norm,one_mul]
-
-theorem unitary_dist_inv_mul_one (a b : SU2) : dist (a⁻¹ * b) 1 = dist a b := by
-  rw [← unitary_dist_left a (a⁻¹ * b) 1]
-  simp [dist_comm]
-
-theorem unitary_dist_mul_one_le (a b : SU2) :
-    dist (a * b) 1 ≤ dist a 1 + dist b 1 := by
-  calc
-    dist (a * b) 1 ≤ dist (a * b) a + dist a 1 := dist_triangle _ _ _
-    _ = dist b 1 + dist a 1 := by
-      exact congrArg (fun t => t + dist a 1) (by simpa only [mul_one] using unitary_dist_left a b 1)
-    _ = dist a 1 + dist b 1 := add_comm _ _
-
-section Connection
-variable {V : Type*} [Fintype V]
-variable (L : SimpleGraph V)
-
-def Related (a b : V) : Prop := a = b ∨ L.Adj a b
-
-omit [Fintype V] in
-theorem Related.refl (a : V) : Related L a a := Or.inl rfl
-
-omit [Fintype V] in
-theorem Related.symm {a b : V} : Related L a b → Related L b a := by
-  rintro (rfl | h)
-  · exact Related.refl L _
-  · exact Or.inr h.symm
-
-def Star (a : V) (b : TrianglePoint V) : Prop :=
-  ∀ v, (b : V → ℝ) v ≠ 0 → Related L a v
-
-def CliquePoint (b : TrianglePoint V) : Prop :=
-  ∀ a, (b : V → ℝ) a ≠ 0 → Star L a b
-
-def Compatible (b c : TrianglePoint V) : Prop :=
-  ∀ v w, ((b : V → ℝ) v ≠ 0 ∨ (c : V → ℝ) v ≠ 0) →
-    ((b : V → ℝ) w ≠ 0 ∨ (c : V → ℝ) w ≠ 0) → Related L v w
-
-def root (b : TrianglePoint V) : V := Classical.choose (point_exists_nonzero b)
-
-theorem root_mem (b : TrianglePoint V) : (b : V → ℝ) (root b) ≠ 0 :=
-  Classical.choose_spec (point_exists_nonzero b)
-
-variable (h : V → V → SU2) (q : V → ℍ)
-
-def FlatConnection : Prop := ∀ a b c, Related L a b → Related L b c →
-  Related L a c → h a b * h b c = h a c
-
-theorem rawGauge_transition (hf : FlatConnection L h) {a c : V}
-    {b : TrianglePoint V} (hac : Related L a c) (ha : Star L a b) (hc : Star L c b) :
-    rawGauge h q a b = (h a c : ℍ) * rawGauge h q c b := by
-  classical
-  simp only [rawGauge,Finset.mul_sum]
-  apply Finset.sum_congr rfl
-  intro v _
-  by_cases hv : (b : V → ℝ) v = 0
-  · simp [hv]
-  · rw [mul_smul_comm,← mul_assoc,← Submonoid.coe_mul,hf a c v hac (hc v hv) (ha v hv)]
-
-theorem gauge_transition (hf : FlatConnection L h) (hq : GeneralPosition h q)
-    {a c : V} {b : TrianglePoint V} (hac : Related L a c)
-    (ha : Star L a b) (hc : Star L c b) :
-    gauge h q a b = h a c * gauge h q c b := by
-  unfold gauge
-  rw [rawGauge_transition L h q hf hac ha hc]
-  exact normalize_left _ (rawGauge_ne_zero h q hq c b)
-
-def chartTransport (a : V) (b c : TrianglePoint V) : SU2 :=
-  (gauge h q a b)⁻¹ * gauge h q a c
-
-def transport (b c : TrianglePoint V) : SU2 := chartTransport h q (root b) b c
-
-theorem transport_chart (hf : FlatConnection L h) (hq : GeneralPosition h q)
-    {b c : TrianglePoint V} (hb : Compatible L b c) {a : V}
-    (hab : Star L a b) (hac : Star L a c) :
-    transport h q b c = chartTransport h q a b c := by
-  have hrb : Star L (root b) b := fun v hv => hb _ _ (Or.inl (root_mem b)) (Or.inl hv)
-  have hrc : Star L (root b) c := fun v hv => hb _ _ (Or.inl (root_mem b)) (Or.inr hv)
-  have har : Related L a (root b) := hab _ (root_mem b)
-  unfold transport chartTransport
-  rw [gauge_transition L h q hf hq har hab hrb,
-    gauge_transition L h q hf hq har hac hrc]
-  group
-
-theorem compatible_symm {b c : TrianglePoint V} (hb : Compatible L b c) :
-    Compatible L c b := fun v w hv hw => hb v w hv.symm hw.symm
-
-theorem transport_reverse (hf : FlatConnection L h) (hq : GeneralPosition h q)
-    {b c : TrianglePoint V} (hb : Compatible L b c) :
-    transport h q c b = (transport h q b c)⁻¹ := by
-  have hrc : Star L (root b) c := fun v hv => hb _ _ (Or.inl (root_mem b)) (Or.inr hv)
-  have hrb : Star L (root b) b := fun v hv => hb _ _ (Or.inl (root_mem b)) (Or.inl hv)
-  rw [transport_chart L h q hf hq (compatible_symm L hb) hrc hrb]
-  unfold transport chartTransport
-  group
-
-theorem chartTransport_trans (a : V) (b c d : TrianglePoint V) :
-    chartTransport h q a b c * chartTransport h q a c d = chartTransport h q a b d := by
-  unfold chartTransport
-  group
-
-theorem exists_large_coordinate (b : TrianglePoint V) :
-    ∃ a, 1 / (Fintype.card V + 1 : ℝ) < (b : V → ℝ) a := by
-  by_contra! H
-  have hs := Finset.sum_le_sum (s := Finset.univ) (fun a _ => H a)
-  have hb := (point_simplex b).2
-  rw [hb,Finset.sum_const,Finset.card_univ,nsmul_eq_mul] at hs
-  have hn : 0 ≤ (Fintype.card V : ℝ) := Nat.cast_nonneg _
-  have hd : 0 < (Fintype.card V : ℝ) + 1 := by positivity
-  have he := (le_div_iff₀ hd).mp (by simpa only [mul_one_div] using hs)
-  linarith
-
-theorem nearby_star {b c : TrianglePoint V} {a : V}
-    (ha : 1 / (Fintype.card V + 1 : ℝ) < (b : V → ℝ) a)
-    (hc : CliquePoint L c) (hbc : dist b c < 1 / (Fintype.card V + 1 : ℝ)) :
-    Star L a c := by
-  apply hc a
-  intro he
-  have hd := lt_of_le_of_lt (dist_le_pi_dist (b : V → ℝ) (c : V → ℝ) a) hbc
-  rw [he, Real.dist_eq,sub_zero,abs_of_nonneg ((point_simplex b).1 a)] at hd
-  exact (not_lt_of_ge ha.le) hd
-
-theorem chartTransport_dist (a : V) (b c : TrianglePoint V) :
-    dist (chartTransport h q a b c) 1 = dist (gauge h q a b) (gauge h q a c) :=
-  unitary_dist_inv_mul_one _ _
-
-theorem transport_uniform (hq : GeneralPosition h q) {ε : ℝ} (hε : 0 < ε) :
-    ∃ δ > 0, ∀ b c : TrianglePoint V, dist b c < δ → dist (transport h q b c) 1 < ε := by
-  obtain ⟨δ,hδ,H⟩ := gauges_uniform h q hq hε
-  exact ⟨δ,hδ,fun b c hbc => (chartTransport_dist h q _ b c).symm ▸ H b c hbc (root b)⟩
-
-/-- Two compatible pieces, rather than a chosen homotopy class of paths, suffice. -/
-def edgeTransport (b c d : TrianglePoint V) : SU2 := transport h q b c * transport h q c d
-
-theorem edgeTransport_chart (hf : FlatConnection L h) (hq : GeneralPosition h q)
-    {b c d : TrianglePoint V} (hbc : Compatible L b c) (hcd : Compatible L c d)
-    {a : V} (hab : Star L a b) (hac : Star L a c) (had : Star L a d) :
-    edgeTransport h q b c d = chartTransport h q a b d := by
-  rw [edgeTransport,transport_chart L h q hf hq hbc hab hac,
-    transport_chart L h q hf hq hcd hac had,chartTransport_trans]
-
-theorem edgeTransport_reverse (hf : FlatConnection L h) (hq : GeneralPosition h q)
-    {b c d : TrianglePoint V} (hbc : Compatible L b c) (hcd : Compatible L c d) :
-    edgeTransport h q d c b = (edgeTransport h q b c d)⁻¹ := by
-  rw [edgeTransport,edgeTransport,transport_reverse L h q hf hq hbc,
-    transport_reverse L h q hf hq hcd,mul_inv_rev]
-
-theorem edgeTransport_uniform (hq : GeneralPosition h q) {ε : ℝ} (hε : 0 < ε) :
-    ∃ δ > 0, ∀ b c d : TrianglePoint V, dist b c < δ → dist c d < δ →
-      dist (edgeTransport h q b c d) 1 < ε := by
-  obtain ⟨δ,hδ,H⟩ := transport_uniform h q hq (show 0 < ε/2 by positivity)
-  refine ⟨δ,hδ,?_⟩
-  intro b c d hbc hcd
-  apply lt_of_le_of_lt (unitary_dist_mul_one_le _ _)
-  have h₁ := H b c hbc
-  have h₂ := H c d hcd
-  linarith
-
-/-- All three edge images in one chart have exactly trivial triangle holonomy. -/
-theorem edgeTransport_triangle (hf : FlatConnection L h) (hq : GeneralPosition h q)
-    (b c d u v w : TrianglePoint V) {a : V}
-    (hbu : Compatible L b u) (huc : Compatible L u c)
-    (hcv : Compatible L c v) (hvd : Compatible L v d)
-    (hdw : Compatible L d w) (hwb : Compatible L w b)
-    (hab : Star L a b) (hac : Star L a c) (had : Star L a d)
-    (hau : Star L a u) (hav : Star L a v) (haw : Star L a w) :
-    edgeTransport h q b u c * edgeTransport h q c v d * edgeTransport h q d w b = 1 := by
-  rw [edgeTransport_chart L h q hf hq hbu huc hab hau hac,
-    edgeTransport_chart L h q hf hq hcv hvd hac hav had,
-    edgeTransport_chart L h q hf hq hdw hwb had haw hab]
-  unfold chartTransport
-  group
-
-/-- A uniform local chart for any finite or infinite family of nearby clique points. -/
-theorem nearby_family_chart {I : Type*} (b : TrianglePoint V) (c : I → TrianglePoint V)
-    (hc : ∀ i, CliquePoint L (c i))
-    (hbc : ∀ i, dist b (c i) < 1 / (Fintype.card V + 1 : ℝ)) :
-    ∃ a, ∀ i, Star L a (c i) := by
-  obtain ⟨a,ha⟩ := exists_large_coordinate b
-  exact ⟨a,fun i => nearby_star L ha (hc i) (hbc i)⟩
-
-end Connection
-
-end AffineTransport
-
-
-
-
-end EilenbergGanea
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.SmallHolonomy
+import OAI.GroupTheory.RightAngledArtin.TraceWords
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set Filter Topology MeasureTheory
+open scoped Quaternion ContDiff
+
+namespace EilenbergGanea
+namespace AffineTransport
+
+/-- A finite family of proper quaternion subspaces cannot cover ℍ. -/
+theorem exists_avoid_subspaces {I : Type*} [Finite I]
+    (p : I → Submodule ℝ ℍ) (hp : ∀ i, p i ≠ ⊤) :
+    ∃ q : ℍ, ∀ i, q ∉ p i := by
+  have ho (i) : IsOpen ((p i : Set ℍ)ᶜ) :=
+    (p i).closed_of_finiteDimensional.isOpen_compl
+  have hd (i) : Dense ((p i : Set ℍ)ᶜ) := by
+    apply interior_eq_empty_iff_dense_compl.mp
+    apply Set.not_nonempty_iff_eq_empty.mp
+    intro h
+    exact hp i ((p i).eq_top_of_nonempty_interior' h)
+  obtain ⟨q,hq⟩ := (dense_iInter_of_isOpen ho hd).nonempty
+  exact ⟨q,Set.mem_iInter.mp hq⟩
+
+/-- Simultaneous general position for finitely many local quaternion gauges.
+Different vertices may use different invertible linear transformations in each
+chart. No flatness or topology is assumed in this finite-dimensional lemma. -/
+theorem exists_triangle_general_position {A V : Type*} [Finite A] [Fintype V]
+    (T : A → V → ℍ ≃ₗ[ℝ] ℍ) :
+    ∃ q : V → ℍ, ∀ a (s : Finset V), s.card ≤ 3 →
+      LinearIndepOn ℝ (fun v => T a v (q v)) (s : Set V) := by
+  classical
+  suffices ∀ U : Finset V, ∃ q : V → ℍ, ∀ a (s : Finset V), s ⊆ U → s.card ≤ 3 →
+      LinearIndepOn ℝ (fun v => T a v (q v)) (s : Set V) by
+    obtain ⟨q,hq⟩ := this Finset.univ
+    exact ⟨q,fun a s h => hq a s (Finset.subset_univ _) h⟩
+  intro U
+  induction U using Finset.induction with
+  | empty =>
+      refine ⟨0,?_⟩
+      intro a s hs _
+      have : s = ∅ := Finset.subset_empty.mp hs
+      subst s
+      simpa only [Finset.coe_empty,Pi.zero_apply] using
+        linearIndepOn_empty ℝ (fun v => T a v (0 : ℍ))
+  | @insert v U hv ih =>
+      obtain ⟨q,hq⟩ := ih
+      let I := A × {s : Finset V // s ⊆ U ∧ s.card ≤ 2}
+      let p : I → Submodule ℝ ℍ := fun i =>
+        (Submodule.span ℝ ((fun w => T i.1 w (q w)) '' (i.2.val : Set V))).comap
+          (T i.1 v).toLinearMap
+      have hp (i : I) : p i ≠ ⊤ := by
+        have hf : Module.finrank ℝ (Submodule.span ℝ
+            ((fun w => T i.1 w (q w)) '' (i.2.val : Set V))) ≤ 2 := by
+          have hh := finrank_span_finset_le_card (R := ℝ) (i.2.val.image (fun w => T i.1 w (q w)))
+          rw [Finset.coe_image] at hh
+          exact hh.trans ((Finset.card_image_le).trans i.2.property.2)
+        have hproper : Submodule.span ℝ ((fun w => T i.1 w (q w)) ''
+            (i.2.val : Set V)) ≠ ⊤ := by
+          apply ne_of_lt (Submodule.lt_top_of_finrank_lt_finrank _)
+          rw [Quaternion.finrank_eq_four]
+          omega
+        intro hh
+        apply hproper
+        apply top_unique
+        intro x _
+        have hm : (T i.1 v).symm x ∈ p i := by rw [hh]; trivial
+        simpa only [p,Submodule.mem_comap,LinearEquiv.coe_coe,
+          LinearEquiv.apply_symm_apply] using hm
+      obtain ⟨z,hz⟩ := exists_avoid_subspaces p hp
+      refine ⟨Function.update q v z,?_⟩
+      intro a s hs hc
+      by_cases hvs : v ∈ s
+      · have hsub : s.erase v ⊆ U := by
+          intro w hw
+          have hh := hs (Finset.mem_of_mem_erase hw)
+          exact (Finset.mem_insert.mp hh).resolve_left (Finset.ne_of_mem_erase hw)
+        have hcard : (s.erase v).card ≤ 2 := by
+          have he := Finset.card_erase_add_one hvs
+          omega
+        have heq : ∀ w ∈ (s.erase v : Finset V), Function.update q v z w = q w := by
+          intro w hw
+          exact Function.update_of_ne (Finset.ne_of_mem_erase hw) _ _
+        have hi : LinearIndepOn ℝ (fun w => T a w (Function.update q v z w))
+            ((s.erase v : Finset V) : Set V) := by
+          apply (hq a (s.erase v) hsub (by omega)).congr
+          intro w hw
+          dsimp only
+          rw [heq w hw]
+        have hn : T a v (Function.update q v z v) ∉ Submodule.span ℝ
+            ((fun w => T a w (Function.update q v z w)) ''
+              ((s.erase v : Finset V) : Set V)) := by
+          rw [Function.update_self]
+          have him : ((fun w => T a w (Function.update q v z w)) ''
+              ((s.erase v : Finset V) : Set V)) =
+              ((fun w => T a w (q w)) '' ((s.erase v : Finset V) : Set V)) := by
+            apply Set.image_congr
+            intro w hw
+            rw [heq w hw]
+          rw [him]
+          exact hz ⟨a,⟨s.erase v,hsub,hcard⟩⟩
+        have ht := (linearIndepOn_insert (by simp : v ∉ ((s.erase v : Finset V) : Set V))).mpr
+          ⟨hi,hn⟩
+        simpa only [← Finset.coe_insert,Finset.insert_erase hvs] using ht
+      · have hsub : s ⊆ U := by
+          intro w hw
+          have hh := hs hw
+          rcases Finset.mem_insert.mp hh with rfl | hh
+          · exact False.elim (hvs hw)
+          · exact hh
+        apply (hq a s hsub hc).congr
+        intro w hw
+        dsimp only
+        rw [Function.update_of_ne (by intro h; subst w; exact hvs hw)]
+
+
+@[simp] theorem unit_norm (h : SU2) : ‖(h : ℍ)‖ = 1 :=
+  (quaternion_mem_unitary_iff _).mp h.property
+
+def leftLinearEquiv (h : SU2) : ℍ ≃ₗ[ℝ] ℍ where
+  toFun q := (h : ℍ) * q
+  invFun q := (h⁻¹ : SU2) * q
+  left_inv q := by simp [← mul_assoc,← Submonoid.coe_mul]
+  right_inv q := by simp [← mul_assoc,← Submonoid.coe_mul]
+  map_add' _ _ := mul_add _ _ _
+  map_smul' r q := mul_smul_comm r (h : ℍ) q
+
+@[simp] theorem leftLinearEquiv_apply (h : SU2) (q : ℍ) :
+    leftLinearEquiv h q = (h : ℍ) * q := rfl
+
+theorem normalize_left (h : SU2) {q : ℍ} (hq : q ≠ 0) :
+    normalizeQuaternion ((h : ℍ) * q) = h * normalizeQuaternion q := by
+  have hh : (h : ℍ) * q ≠ 0 := by
+    exact (leftLinearEquiv h).map_ne_zero_iff.mpr hq
+  apply Subtype.ext
+  rw [normalizeQuaternion_coe_of_ne_zero hh]
+  change ‖(h : ℍ) * q‖⁻¹ • ((h : ℍ) * q) = (h : ℍ) * (normalizeQuaternion q : ℍ)
+  rw [normalizeQuaternion_coe_of_ne_zero hq,norm_mul,unit_norm,one_mul,mul_smul_comm]
+
+section FiniteSimplex
+variable {V : Type*} [Fintype V]
+
+def coordinateSimplex : Set (V → ℝ) :=
+  {weights | (∀ vertex, 0 ≤ weights vertex) ∧ ∑ vertex, weights vertex = 1}
+
+/-- Closed coordinate face of the standard simplex. -/
+def face (s : Finset V) : Set (V → ℝ) :=
+  {b | b ∈ coordinateSimplex ∧ ∀ v ∉ s, b v = 0}
+
+/-- The whole two-skeleton of a finite standard simplex. -/
+def triangleSet : Set (V → ℝ) :=
+  ⋃ s : {s : Finset V // s.card ≤ 3}, face s.val
+
+abbrev TrianglePoint (V : Type*) [Fintype V] := (triangleSet : Set (V → ℝ))
+
+theorem mem_triangleSet {b : V → ℝ} : b ∈ triangleSet ↔
+    ∃ s : Finset V, s.card ≤ 3 ∧ b ∈ coordinateSimplex ∧ ∀ v ∉ s, b v = 0 := by
+  simp only [triangleSet,Set.mem_iUnion,face,Set.mem_ofPred_eq,Subtype.exists,exists_prop]
+
+omit [Fintype V] in
+private theorem isClosed_face_support (s : Finset V) :
+    IsClosed {b : V → ℝ | ∀ v ∉ s, b v = 0} := by
+  have he : {b : V → ℝ | ∀ v ∉ s, b v = 0} =
+      ⋂ v, ⋂ (_ : v ∉ s), {b : V → ℝ | b v = 0} := by ext b; simp
+  rw [he]
+  exact isClosed_iInter fun v => isClosed_iInter fun _ =>
+    isClosed_eq (continuous_apply v) continuous_const
+
+theorem isCompact_face (s : Finset V) : IsCompact (face s) := by
+  have compact_simplex : IsCompact (coordinateSimplex : Set (V → ℝ)) := by
+    convert isCompact_range
+      (Convexity.StdSimplex.isEmbedding_toFun_comp_weights ℝ V).continuous using 1
+    rw [Convexity.StdSimplex.range_toFun_comp_weights]
+    ext weights
+    simp [coordinateSimplex]
+  exact compact_simplex.inter_right (isClosed_face_support s)
+
+theorem isCompact_triangleSet : IsCompact (triangleSet : Set (V → ℝ)) :=
+  isCompact_iUnion fun _ => isCompact_face _
+
+instance : CompactSpace (TrianglePoint V) :=
+  isCompact_iff_compactSpace.mp isCompact_triangleSet
+
+theorem point_simplex (b : TrianglePoint V) : (b : V → ℝ) ∈ coordinateSimplex := by
+  obtain ⟨_,_,h,_⟩ := mem_triangleSet.mp b.property
+  exact h
+
+theorem point_exists_nonzero (b : TrianglePoint V) : ∃ v, (b : V → ℝ) v ≠ 0 := by
+  by_contra! h
+  have he := (point_simplex b).2
+  simp only [h,Finset.sum_const_zero] at he
+  norm_num at he
+
+variable (h : V → V → SU2) (q : V → ℍ)
+
+def rawGauge (a : V) (b : V → ℝ) : ℍ := ∑ v, b v • ((h a v : ℍ) * q v)
+
+def GeneralPosition : Prop := ∀ a (s : Finset V), s.card ≤ 3 →
+  LinearIndepOn ℝ (fun v => (h a v : ℍ) * q v) (s : Set V)
+
+theorem exists_generalPosition : ∃ q : V → ℍ, GeneralPosition h q := by
+  obtain ⟨q,hq⟩ := exists_triangle_general_position (fun a v => leftLinearEquiv (h a v))
+  exact ⟨q,hq⟩
+
+theorem rawGauge_ne_zero (hq : GeneralPosition h q) (a : V) (b : TrianglePoint V) :
+    rawGauge h q a b ≠ 0 := by
+  classical
+  obtain ⟨s,hs,hb,hout⟩ := mem_triangleSet.mp b.property
+  intro he
+  have he' : ∑ v ∈ s, (b : V → ℝ) v • ((h a v : ℍ) * q v) = 0 := by
+    rw [← he]
+    apply Finset.sum_subset (Finset.subset_univ _)
+    intro v _ hv
+    rw [hout v hv,zero_smul]
+  have hz := (linearIndepOn_iff'.mp (hq a s hs)) s b (Set.Subset.refl _) he'
+  have hall : ∀ v, (b : V → ℝ) v = 0 := by
+    intro v
+    by_cases hv : v ∈ s
+    · exact hz v hv
+    · exact hout v hv
+  have heq := hb.2
+  simp only [hall,Finset.sum_const_zero] at heq
+  norm_num at heq
+
+def gauge (a : V) (b : TrianglePoint V) : SU2 := normalizeQuaternion (rawGauge h q a b)
+
+theorem continuous_rawGauge (a : V) : Continuous (rawGauge h q a) :=
+  continuous_finsetSum _ fun v _ => (continuous_apply v).smul continuous_const
+
+theorem continuous_gauge (hq : GeneralPosition h q) (a : V) : Continuous (gauge h q a) := by
+  apply continuous_iff_continuousAt.mpr
+  intro b
+  exact ContinuousAt.comp (f := fun c : TrianglePoint V => rawGauge h q a c)
+    (normalizeQuaternion_continuousAt (rawGauge_ne_zero h q hq a b))
+    ((continuous_rawGauge h q a).comp continuous_subtype_val).continuousAt
+
+/-- Uniform modulus over all charts, including across the edges of all simplices. -/
+theorem gauges_uniform (hq : GeneralPosition h q) {ε : ℝ} (hε : 0 < ε) :
+    ∃ δ > 0, ∀ b c : TrianglePoint V, dist b c < δ →
+      ∀ a, dist (gauge h q a b) (gauge h q a c) < ε := by
+  have hc : Continuous (fun b : TrianglePoint V => fun a => gauge h q a b) :=
+    continuous_pi fun a => continuous_gauge h q hq a
+  have hu := CompactSpace.uniformContinuous_of_continuous hc
+  obtain ⟨δ,hδ,H⟩ := Metric.uniformContinuous_iff.mp hu ε hε
+  refine ⟨δ,hδ,?_⟩
+  intro b c hbc a
+  exact lt_of_le_of_lt (dist_le_pi_dist (fun a => gauge h q a b) (fun a => gauge h q a c) a) (H hbc)
+
+end FiniteSimplex
+
+theorem unitary_dist_left (a b c : SU2) : dist (a * b) (a * c) = dist b c := by
+  change dist ((a : ℍ) * (b : ℍ)) ((a : ℍ) * (c : ℍ)) = dist (b : ℍ) (c : ℍ)
+  rw [dist_eq_norm,dist_eq_norm,← mul_sub,norm_mul,unit_norm,one_mul]
+
+theorem unitary_dist_inv_mul_one (a b : SU2) : dist (a⁻¹ * b) 1 = dist a b := by
+  rw [← unitary_dist_left a (a⁻¹ * b) 1]
+  simp [dist_comm]
+
+theorem unitary_dist_mul_one_le (a b : SU2) :
+    dist (a * b) 1 ≤ dist a 1 + dist b 1 := by
+  calc
+    dist (a * b) 1 ≤ dist (a * b) a + dist a 1 := dist_triangle _ _ _
+    _ = dist b 1 + dist a 1 := by
+      exact congrArg (fun t => t + dist a 1) (by simpa only [mul_one] using unitary_dist_left a b 1)
+    _ = dist a 1 + dist b 1 := add_comm _ _
+
+section Connection
+variable {V : Type*} [Fintype V]
+variable (L : SimpleGraph V)
+
+def Related (a b : V) : Prop := a = b ∨ L.Adj a b
+
+omit [Fintype V] in
+theorem Related.refl (a : V) : Related L a a := Or.inl rfl
+
+omit [Fintype V] in
+theorem Related.symm {a b : V} : Related L a b → Related L b a := by
+  rintro (rfl | h)
+  · exact Related.refl L _
+  · exact Or.inr h.symm
+
+def Star (a : V) (b : TrianglePoint V) : Prop :=
+  ∀ v, (b : V → ℝ) v ≠ 0 → Related L a v
+
+def CliquePoint (b : TrianglePoint V) : Prop :=
+  ∀ a, (b : V → ℝ) a ≠ 0 → Star L a b
+
+def Compatible (b c : TrianglePoint V) : Prop :=
+  ∀ v w, ((b : V → ℝ) v ≠ 0 ∨ (c : V → ℝ) v ≠ 0) →
+    ((b : V → ℝ) w ≠ 0 ∨ (c : V → ℝ) w ≠ 0) → Related L v w
+
+def root (b : TrianglePoint V) : V := Classical.choose (point_exists_nonzero b)
+
+theorem root_mem (b : TrianglePoint V) : (b : V → ℝ) (root b) ≠ 0 :=
+  Classical.choose_spec (point_exists_nonzero b)
+
+variable (h : V → V → SU2) (q : V → ℍ)
+
+def FlatConnection : Prop := ∀ a b c, Related L a b → Related L b c →
+  Related L a c → h a b * h b c = h a c
+
+theorem rawGauge_transition (hf : FlatConnection L h) {a c : V}
+    {b : TrianglePoint V} (hac : Related L a c) (ha : Star L a b) (hc : Star L c b) :
+    rawGauge h q a b = (h a c : ℍ) * rawGauge h q c b := by
+  classical
+  simp only [rawGauge,Finset.mul_sum]
+  apply Finset.sum_congr rfl
+  intro v _
+  by_cases hv : (b : V → ℝ) v = 0
+  · simp [hv]
+  · rw [mul_smul_comm,← mul_assoc,← Submonoid.coe_mul,hf a c v hac (hc v hv) (ha v hv)]
+
+theorem gauge_transition (hf : FlatConnection L h) (hq : GeneralPosition h q)
+    {a c : V} {b : TrianglePoint V} (hac : Related L a c)
+    (ha : Star L a b) (hc : Star L c b) :
+    gauge h q a b = h a c * gauge h q c b := by
+  unfold gauge
+  rw [rawGauge_transition L h q hf hac ha hc]
+  exact normalize_left _ (rawGauge_ne_zero h q hq c b)
+
+def chartTransport (a : V) (b c : TrianglePoint V) : SU2 :=
+  (gauge h q a b)⁻¹ * gauge h q a c
+
+def transport (b c : TrianglePoint V) : SU2 := chartTransport h q (root b) b c
+
+theorem transport_chart (hf : FlatConnection L h) (hq : GeneralPosition h q)
+    {b c : TrianglePoint V} (hb : Compatible L b c) {a : V}
+    (hab : Star L a b) (hac : Star L a c) :
+    transport h q b c = chartTransport h q a b c := by
+  have hrb : Star L (root b) b := fun v hv => hb _ _ (Or.inl (root_mem b)) (Or.inl hv)
+  have hrc : Star L (root b) c := fun v hv => hb _ _ (Or.inl (root_mem b)) (Or.inr hv)
+  have har : Related L a (root b) := hab _ (root_mem b)
+  unfold transport chartTransport
+  rw [gauge_transition L h q hf hq har hab hrb,
+    gauge_transition L h q hf hq har hac hrc]
+  group
+
+theorem compatible_symm {b c : TrianglePoint V} (hb : Compatible L b c) :
+    Compatible L c b := fun v w hv hw => hb v w hv.symm hw.symm
+
+theorem transport_reverse (hf : FlatConnection L h) (hq : GeneralPosition h q)
+    {b c : TrianglePoint V} (hb : Compatible L b c) :
+    transport h q c b = (transport h q b c)⁻¹ := by
+  have hrc : Star L (root b) c := fun v hv => hb _ _ (Or.inl (root_mem b)) (Or.inr hv)
+  have hrb : Star L (root b) b := fun v hv => hb _ _ (Or.inl (root_mem b)) (Or.inl hv)
+  rw [transport_chart L h q hf hq (compatible_symm L hb) hrc hrb]
+  unfold transport chartTransport
+  group
+
+theorem chartTransport_trans (a : V) (b c d : TrianglePoint V) :
+    chartTransport h q a b c * chartTransport h q a c d = chartTransport h q a b d := by
+  unfold chartTransport
+  group
+
+theorem exists_large_coordinate (b : TrianglePoint V) :
+    ∃ a, 1 / (Fintype.card V + 1 : ℝ) < (b : V → ℝ) a := by
+  by_contra! H
+  have hs := Finset.sum_le_sum (s := Finset.univ) (fun a _ => H a)
+  have hb := (point_simplex b).2
+  rw [hb,Finset.sum_const,Finset.card_univ,nsmul_eq_mul] at hs
+  have hn : 0 ≤ (Fintype.card V : ℝ) := Nat.cast_nonneg _
+  have hd : 0 < (Fintype.card V : ℝ) + 1 := by positivity
+  have he := (le_div_iff₀ hd).mp (by simpa only [mul_one_div] using hs)
+  linarith
+
+theorem nearby_star {b c : TrianglePoint V} {a : V}
+    (ha : 1 / (Fintype.card V + 1 : ℝ) < (b : V → ℝ) a)
+    (hc : CliquePoint L c) (hbc : dist b c < 1 / (Fintype.card V + 1 : ℝ)) :
+    Star L a c := by
+  apply hc a
+  intro he
+  have hd := lt_of_le_of_lt (dist_le_pi_dist (b : V → ℝ) (c : V → ℝ) a) hbc
+  rw [he, Real.dist_eq,sub_zero,abs_of_nonneg ((point_simplex b).1 a)] at hd
+  exact (not_lt_of_ge ha.le) hd
+
+theorem chartTransport_dist (a : V) (b c : TrianglePoint V) :
+    dist (chartTransport h q a b c) 1 = dist (gauge h q a b) (gauge h q a c) :=
+  unitary_dist_inv_mul_one _ _
+
+theorem transport_uniform (hq : GeneralPosition h q) {ε : ℝ} (hε : 0 < ε) :
+    ∃ δ > 0, ∀ b c : TrianglePoint V, dist b c < δ → dist (transport h q b c) 1 < ε := by
+  obtain ⟨δ,hδ,H⟩ := gauges_uniform h q hq hε
+  exact ⟨δ,hδ,fun b c hbc => (chartTransport_dist h q _ b c).symm ▸ H b c hbc (root b)⟩
+
+/-- Two compatible pieces, rather than a chosen homotopy class of paths, suffice. -/
+def edgeTransport (b c d : TrianglePoint V) : SU2 := transport h q b c * transport h q c d
+
+theorem edgeTransport_chart (hf : FlatConnection L h) (hq : GeneralPosition h q)
+    {b c d : TrianglePoint V} (hbc : Compatible L b c) (hcd : Compatible L c d)
+    {a : V} (hab : Star L a b) (hac : Star L a c) (had : Star L a d) :
+    edgeTransport h q b c d = chartTransport h q a b d := by
+  rw [edgeTransport,transport_chart L h q hf hq hbc hab hac,
+    transport_chart L h q hf hq hcd hac had,chartTransport_trans]
+
+theorem edgeTransport_reverse (hf : FlatConnection L h) (hq : GeneralPosition h q)
+    {b c d : TrianglePoint V} (hbc : Compatible L b c) (hcd : Compatible L c d) :
+    edgeTransport h q d c b = (edgeTransport h q b c d)⁻¹ := by
+  rw [edgeTransport,edgeTransport,transport_reverse L h q hf hq hbc,
+    transport_reverse L h q hf hq hcd,mul_inv_rev]
+
+theorem edgeTransport_uniform (hq : GeneralPosition h q) {ε : ℝ} (hε : 0 < ε) :
+    ∃ δ > 0, ∀ b c d : TrianglePoint V, dist b c < δ → dist c d < δ →
+      dist (edgeTransport h q b c d) 1 < ε := by
+  obtain ⟨δ,hδ,H⟩ := transport_uniform h q hq (show 0 < ε/2 by positivity)
+  refine ⟨δ,hδ,?_⟩
+  intro b c d hbc hcd
+  apply lt_of_le_of_lt (unitary_dist_mul_one_le _ _)
+  have h₁ := H b c hbc
+  have h₂ := H c d hcd
+  linarith
+
+/-- All three edge images in one chart have exactly trivial triangle holonomy. -/
+theorem edgeTransport_triangle (hf : FlatConnection L h) (hq : GeneralPosition h q)
+    (b c d u v w : TrianglePoint V) {a : V}
+    (hbu : Compatible L b u) (huc : Compatible L u c)
+    (hcv : Compatible L c v) (hvd : Compatible L v d)
+    (hdw : Compatible L d w) (hwb : Compatible L w b)
+    (hab : Star L a b) (hac : Star L a c) (had : Star L a d)
+    (hau : Star L a u) (hav : Star L a v) (haw : Star L a w) :
+    edgeTransport h q b u c * edgeTransport h q c v d * edgeTransport h q d w b = 1 := by
+  rw [edgeTransport_chart L h q hf hq hbu huc hab hau hac,
+    edgeTransport_chart L h q hf hq hcv hvd hac hav had,
+    edgeTransport_chart L h q hf hq hdw hwb had haw hab]
+  unfold chartTransport
+  group
+
+/-- A uniform local chart for any finite or infinite family of nearby clique points. -/
+theorem nearby_family_chart {I : Type*} (b : TrianglePoint V) (c : I → TrianglePoint V)
+    (hc : ∀ i, CliquePoint L (c i))
+    (hbc : ∀ i, dist b (c i) < 1 / (Fintype.card V + 1 : ℝ)) :
+    ∃ a, ∀ i, Star L a (c i) := by
+  obtain ⟨a,ha⟩ := exists_large_coordinate b
+  exact ⟨a,fun i => nearby_star L ha (hc i) (hbc i)⟩
+
+end Connection
+
+end AffineTransport
+
+
+
+
+end EilenbergGanea
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/CliqueCohomology.lean

@@ -1,232 +1,232 @@
-import OAI.Topology.EilenbergGanea.Model
-import Mathlib.CategoryTheory.Abelian.Projective.Ext
-import Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic
-import Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree
-import Mathlib.RepresentationTheory.Homological.GroupCohomology.LongExactSequence
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set
-
-namespace EilenbergGanea
-section CliqueSquare
-variable {V : Type*} (L : SimpleGraph V)
-
- def generatorExponent (v : V) : ArtinGroup L →* Multiplicative ℤ := by
-  classical
-  exact artinLift L (fun w => Multiplicative.ofAdd (if w = v then 1 else 0))
-    (fun _ _ _ => Commute.all _ _)
-
-@[simp] theorem generatorExponent_of [DecidableEq V] (v w : V) :
-    generatorExponent L v (artinGenerator L w) = Multiplicative.ofAdd (if w = v then 1 else 0) := by
-  classical
-  simp [generatorExponent]
-
- def heightDifference (v b : V) : (height L).ker :=
-  ⟨artinGenerator L v * (artinGenerator L b)⁻¹, by simp⟩
-
- theorem heightDifference_commute (v w b : V)
-    (hvw : L.Adj v w) (hvb : L.Adj v b) (hwb : L.Adj w b) :
-    Commute (heightDifference L v b) (heightDifference L w b) := by
-  apply Subtype.ext
-  exact (((adjacent_generators_commute L hvw).mul_right
-    (adjacent_generators_commute L hvb).inv_right).mul_left
-      (((adjacent_generators_commute L hwb).symm.inv_left).mul_right (Commute.refl _)))
-
- def cliqueSquare (v w b : V) (hvw : L.Adj v w) (hvb : L.Adj v b) (hwb : L.Adj w b) :
-    Multiplicative (ℤ × ℤ) →* (height L).ker where
-  toFun n := heightDifference L v b ^ n.toAdd.1 * heightDifference L w b ^ n.toAdd.2
-  map_one' := by simp
-  map_mul' n m := by
-    change _ ^ (n.toAdd.1 + m.toAdd.1) * _ ^ (n.toAdd.2 + m.toAdd.2) = _
-    rw [zpow_add, zpow_add]
-    simp only [mul_assoc]
-    congr 1
-    rw [← mul_assoc, ((heightDifference_commute L v w b hvw hvb hwb).zpow_zpow m.toAdd.1 n.toAdd.2).eq, mul_assoc]
-
- theorem cliqueSquare_injective (v w b : V) (hvw : L.Adj v w) (hvb : L.Adj v b) (hwb : L.Adj w b) :
-    Function.Injective (cliqueSquare L v w b hvw hvb hwb) := by
-  classical
-  have hvw' : v ≠ w := L.ne_of_adj hvw
-  have hvb' : v ≠ b := L.ne_of_adj hvb
-  have hwb' : w ≠ b := L.ne_of_adj hwb
-  intro n m he
-  have h₁ := congrArg (fun x : (height L).ker => generatorExponent L v x.val) he
-  have h₂ := congrArg (fun x : (height L).ker => generatorExponent L w x.val) he
-  have hn₁ : n.toAdd.1 = m.toAdd.1 := by
-    simpa [cliqueSquare, heightDifference, generatorExponent_of, hvw', hvw'.symm, hvb', hvb'.symm] using congrArg Multiplicative.toAdd h₁
-  have hn₂ : n.toAdd.2 = m.toAdd.2 := by
-    simpa [cliqueSquare, heightDifference, generatorExponent_of, hvw', hwb', hwb'.symm] using congrArg Multiplicative.toAdd h₂
-  exact congrArg Multiplicative.ofAdd (Prod.ext hn₁ hn₂)
-end CliqueSquare
-
- def seedCliqueVertex : SeedCell := .inl (.inl none)
- def seedCliqueEdge : SeedCell := .inr (.inl (.inl (false, 0)))
- def seedCliqueFace : SeedCell := .inr (.inr ⟨false, ⟨0, by decide⟩⟩)
-
- theorem source_contains_int_square :
-    ∃ f : Multiplicative (ℤ × ℤ) →* SourceGroup, Function.Injective f := by
-  have hve : seedGraph.Adj seedCliqueVertex seedCliqueEdge := by
-    change properFace _ _ ∨ _
-    left
-    simp [properFace, seedCliqueVertex, seedCliqueEdge, edgeVertices, circleStart]
-  have hvf : seedGraph.Adj seedCliqueVertex seedCliqueFace := by
-    change properFace _ _ ∨ _
-    left
-    simp [properFace, seedCliqueVertex, seedCliqueFace, sectorVertices, sectorEdges, edgeVertices, sectorCircleEdge, sectorLetter, seedWord, circleStart, circleEnd]
-  have hef : seedGraph.Adj seedCliqueEdge seedCliqueFace := by
-    change properFace _ _ ∨ _
-    left
-    simp [properFace, seedCliqueEdge, seedCliqueFace, sectorEdges, sectorCircleEdge, sectorLetter, seedWord]
-  exact ⟨cliqueSquare seedGraph _ _ _ hve hvf hef,
-    cliqueSquare_injective seedGraph _ _ _ hve hvf hef⟩
-
-section CupCocycle
-open groupCohomology
-variable {G : Type} [Group G]
-
-/-- The cup product of two integral exponent homomorphisms. -/
-def exponentCup (a b : G →* Multiplicative ℤ) : cocycles₂ (Rep.trivial ℤ G ℤ) :=
-  ⟨fun p => (a p.1).toAdd * (b p.2).toAdd, by
-    rw [mem_cocycles₂_iff]
-    intro g h j
-    simp only [map_mul, toAdd_mul]
-    change ((a g).toAdd + (a h).toAdd) * (b j).toAdd + (a g).toAdd * (b h).toAdd =
-      (a h).toAdd * (b j).toAdd + (a g).toAdd * ((b h).toAdd + (b j).toAdd)
-    ring⟩
-
-/-- Antisymmetrizing on a commuting pair detects a nonzero integral class. -/
-theorem exponentCup_not_coboundary (a b : G →* Multiplicative ℤ) (g h : G)
-    (hc : Commute g h)
-    (hag : (a g).toAdd = 1) (hah : (a h).toAdd = 0)
-    (hbg : (b g).toAdd = 0) (hbh : (b h).toAdd = 1) :
-    ⇑(exponentCup a b) ∉ coboundaries₂ (Rep.trivial ℤ G ℤ) := by
-  rintro ⟨f, hf⟩
-  have h₁ := congrFun hf (g, h)
-  have h₂ := congrFun hf (h, g)
-  change f h - f (g*h) + f g = (a g).toAdd * (b h).toAdd at h₁
-  change f g - f (h*g) + f h = (a h).toAdd * (b g).toAdd at h₂
-  rw [hag, hbh] at h₁
-  rw [hah, hbg, ← hc.eq] at h₂
-  change (f h : ℤ) - f (g*h) + f g = 1 * 1 at h₁
-  change (f g : ℤ) - f (g*h) + f h = 0 * 0 at h₂
-  linarith
-
-theorem exponentCup_class_ne_zero (a b : G →* Multiplicative ℤ) (g h : G)
-    (hc : Commute g h)
-    (hag : (a g).toAdd = 1) (hah : (a h).toAdd = 0)
-    (hbg : (b g).toAdd = 0) (hbh : (b h).toAdd = 1) :
-    H2π _ (exponentCup a b) ≠ 0 := by
-  intro he
-  exact exponentCup_not_coboundary a b g h hc hag hah hbg hbh
-    ((H2π_eq_zero_iff (A := Rep.trivial ℤ G ℤ) (exponentCup a b)).mp he)
-
-end CupCocycle
-
-section CliqueCohomology
-open groupCohomology
-variable {V : Type} (L : SimpleGraph V)
-
-theorem clique_exponent_class_nonzero (v w b : V)
-    (hvw : L.Adj v w) (hvb : L.Adj v b) (hwb : L.Adj w b) :
-    ∃ x : H2 (Rep.trivial ℤ (height L).ker ℤ), x ≠ 0 := by
-  classical
-  let a := (generatorExponent L v).comp (height L).ker.subtype
-  let d := (generatorExponent L w).comp (height L).ker.subtype
-  have hvw' := L.ne_of_adj hvw
-  have hvb' := L.ne_of_adj hvb
-  have hwb' := L.ne_of_adj hwb
-  refine ⟨H2π _ (exponentCup a d), exponentCup_class_ne_zero a d
-    (heightDifference L v b) (heightDifference L w b)
-    (heightDifference_commute L v w b hvw hvb hwb) ?_ ?_ ?_ ?_⟩
-  all_goals simp [a, d, heightDifference, hvw', hvw'.symm, hvb'.symm, hwb'.symm]
-
-end CliqueCohomology
-
-
-open CategoryTheory CategoryTheory.Limits
-
-section
-variable {C D : Type*} [Category* C] [Category* D] [Abelian C] [Abelian D]
-    {X Y : C} (P : ProjectiveResolution X) (F : C ⥤ D)
-    [F.Additive] [F.Full] [F.Faithful]
-    [F.PreservesProjectiveObjects] [F.PreservesHomology]
-
-/-- Vanishing in the literal projective-dimension sense forces every
-2-cocycle in a projective resolution to be a coboundary; the fully faithful
-exact functor permits us to use this for the representation/group-ring equivalence. -/
-theorem cocycle_boundary_of_projectiveDimension_lt_two
-    [HasProjectiveDimensionLT (F.obj X) 2]
-    (f : P.complex.X 2 ⟶ Y) (hf : P.complex.d 3 2 ≫ f = 0) :
-    ∃ g : P.complex.X 1 ⟶ Y, P.complex.d 2 1 ≫ g = f := by
-  let := HasExt.standard D
-  let Q := F.mapProjectiveResolution P
-  have hmf : Q.complex.d 3 2 ≫ F.map f = 0 := by
-    change F.map (P.complex.d 3 2) ≫ F.map f = 0
-    rw [← F.map_comp, hf, F.map_zero]
-  have hc := Abelian.Ext.eq_zero_of_hasProjectiveDimensionLT (Q.extMk (F.map f) 3 rfl hmf) 2 (by rfl)
-  obtain ⟨g, hg⟩ := (Q.extMk_eq_zero_iff (F.map f) 3 rfl hmf 1 rfl).mp hc
-  change F.obj (P.complex.X 1) ⟶ F.obj Y at g
-  change F.map (P.complex.d 2 1) ≫ g = F.map f at hg
-  refine ⟨F.preimage g, F.map_injective ?_⟩
-  rw [F.map_comp, F.map_preimage]
-  exact hg
-
-/-- The same vanishing expressed in the library's older linear-Yoneda model
-of Ext. This theorem proves rather than assumes the bridge between the two APIs. -/
-theorem isZero_homology_two_of_projectiveDimension_lt_two
-    (R : Type*) [Ring R] [Linear R C]
-    [HasProjectiveDimensionLT (F.obj X) 2] :
-    IsZero ((P.complex.linearYonedaObj R Y).homology 2) := by
-  rw [← HomologicalComplex.exactAt_iff_isZero_homology]
-  rw [HomologicalComplex.exactAt_iff' _ 1 2 3 (by simp) (by simp)]
-  rw [ShortComplex.moduleCat_exact_iff]
-  intro f hf
-  change P.complex.d 3 2 ≫ f = 0 at hf
-  exact cocycle_boundary_of_projectiveDimension_lt_two P F f hf
-end
-
-open CategoryTheory CategoryTheory.Limits
-
-theorem two_le_cohomologicalDimension_of_H2_nonzero (G : Type) [Group G]
-    (hx : ∃ x : groupCohomology.H2 (Rep.trivial ℤ G ℤ), x ≠ 0) :
-    2 ≤ cohomologicalDimension G := by
-  change (2 : ℕ) ≤ CategoryTheory.projectiveDimension
-    (ModuleCat.of (MonoidAlgebra ℤ G) (Representation.trivial ℤ G ℤ).asModule)
-  rw [CategoryTheory.projectiveDimension_ge_iff]
-  intro hd
-  let F := Rep.toModuleMonoidAlgebra (k := ℤ) (G := G)
-  let P := Rep.barResolution ℤ G
-  let : F.Additive := Functor.additive_of_preserves_binary_products F
-  have : HasProjectiveDimensionLT (F.obj (Rep.trivial ℤ G ℤ)) 2 := hd
-  have hz := isZero_homology_two_of_projectiveDimension_lt_two
-    (Y := Rep.trivial ℤ G ℤ) P F ℤ
-  have hz' : IsZero (groupCohomology.H2 (Rep.trivial ℤ G ℤ)) :=
-    hz.of_iso (groupCohomologyIso (Rep.trivial ℤ G ℤ) 2 P)
-  obtain ⟨x, hx⟩ := hx
-  exact hx ((ModuleCat.isZero_iff_subsingleton.mp hz').elim x 0)
-
-theorem height_kernel_two_le_cohomologicalDimension {V : Type} (L : SimpleGraph V)
-    (v w b : V) (hvw : L.Adj v w) (hvb : L.Adj v b) (hwb : L.Adj w b) :
-    2 ≤ cohomologicalDimension (height L).ker :=
-  two_le_cohomologicalDimension_of_H2_nonzero _ (clique_exponent_class_nonzero L v w b hvw hvb hwb)
-
-theorem source_two_le_cohomologicalDimension : 2 ≤ cohomologicalDimension SourceGroup := by
-  apply height_kernel_two_le_cohomologicalDimension seedGraph seedCliqueVertex seedCliqueEdge seedCliqueFace
-  · change properFace _ _ ∨ _
-    left
-    simp [properFace, seedCliqueVertex, seedCliqueEdge, edgeVertices, circleStart]
-  · change properFace _ _ ∨ _
-    left
-    simp [properFace, seedCliqueVertex, seedCliqueFace, sectorVertices, sectorEdges, edgeVertices, sectorCircleEdge, sectorLetter, seedWord, circleStart, circleEnd]
-  · change properFace _ _ ∨ _
-    left
-    simp [properFace, seedCliqueEdge, seedCliqueFace, sectorEdges, sectorCircleEdge, sectorLetter, seedWord]
-end EilenbergGanea
-
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.Model
+import Mathlib.CategoryTheory.Abelian.Projective.Ext
+import Mathlib.RepresentationTheory.Homological.GroupCohomology.Basic
+import Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree
+import Mathlib.RepresentationTheory.Homological.GroupCohomology.LongExactSequence
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set
+
+namespace EilenbergGanea
+section CliqueSquare
+variable {V : Type*} (L : SimpleGraph V)
+
+ def generatorExponent (v : V) : ArtinGroup L →* Multiplicative ℤ := by
+  classical
+  exact artinLift L (fun w => Multiplicative.ofAdd (if w = v then 1 else 0))
+    (fun _ _ _ => Commute.all _ _)
+
+@[simp] theorem generatorExponent_of [DecidableEq V] (v w : V) :
+    generatorExponent L v (artinGenerator L w) = Multiplicative.ofAdd (if w = v then 1 else 0) := by
+  classical
+  simp [generatorExponent]
+
+ def heightDifference (v b : V) : (height L).ker :=
+  ⟨artinGenerator L v * (artinGenerator L b)⁻¹, by simp⟩
+
+ theorem heightDifference_commute (v w b : V)
+    (hvw : L.Adj v w) (hvb : L.Adj v b) (hwb : L.Adj w b) :
+    Commute (heightDifference L v b) (heightDifference L w b) := by
+  apply Subtype.ext
+  exact (((adjacent_generators_commute L hvw).mul_right
+    (adjacent_generators_commute L hvb).inv_right).mul_left
+      (((adjacent_generators_commute L hwb).symm.inv_left).mul_right (Commute.refl _)))
+
+ def cliqueSquare (v w b : V) (hvw : L.Adj v w) (hvb : L.Adj v b) (hwb : L.Adj w b) :
+    Multiplicative (ℤ × ℤ) →* (height L).ker where
+  toFun n := heightDifference L v b ^ n.toAdd.1 * heightDifference L w b ^ n.toAdd.2
+  map_one' := by simp
+  map_mul' n m := by
+    change _ ^ (n.toAdd.1 + m.toAdd.1) * _ ^ (n.toAdd.2 + m.toAdd.2) = _
+    rw [zpow_add, zpow_add]
+    simp only [mul_assoc]
+    congr 1
+    rw [← mul_assoc, ((heightDifference_commute L v w b hvw hvb hwb).zpow_zpow m.toAdd.1 n.toAdd.2).eq, mul_assoc]
+
+ theorem cliqueSquare_injective (v w b : V) (hvw : L.Adj v w) (hvb : L.Adj v b) (hwb : L.Adj w b) :
+    Function.Injective (cliqueSquare L v w b hvw hvb hwb) := by
+  classical
+  have hvw' : v ≠ w := L.ne_of_adj hvw
+  have hvb' : v ≠ b := L.ne_of_adj hvb
+  have hwb' : w ≠ b := L.ne_of_adj hwb
+  intro n m he
+  have h₁ := congrArg (fun x : (height L).ker => generatorExponent L v x.val) he
+  have h₂ := congrArg (fun x : (height L).ker => generatorExponent L w x.val) he
+  have hn₁ : n.toAdd.1 = m.toAdd.1 := by
+    simpa [cliqueSquare, heightDifference, generatorExponent_of, hvw', hvw'.symm, hvb', hvb'.symm] using congrArg Multiplicative.toAdd h₁
+  have hn₂ : n.toAdd.2 = m.toAdd.2 := by
+    simpa [cliqueSquare, heightDifference, generatorExponent_of, hvw', hwb', hwb'.symm] using congrArg Multiplicative.toAdd h₂
+  exact congrArg Multiplicative.ofAdd (Prod.ext hn₁ hn₂)
+end CliqueSquare
+
+ def seedCliqueVertex : SeedCell := .inl (.inl none)
+ def seedCliqueEdge : SeedCell := .inr (.inl (.inl (false, 0)))
+ def seedCliqueFace : SeedCell := .inr (.inr ⟨false, ⟨0, by decide⟩⟩)
+
+ theorem source_contains_int_square :
+    ∃ f : Multiplicative (ℤ × ℤ) →* SourceGroup, Function.Injective f := by
+  have hve : seedGraph.Adj seedCliqueVertex seedCliqueEdge := by
+    change properFace _ _ ∨ _
+    left
+    simp [properFace, seedCliqueVertex, seedCliqueEdge, edgeVertices, circleStart]
+  have hvf : seedGraph.Adj seedCliqueVertex seedCliqueFace := by
+    change properFace _ _ ∨ _
+    left
+    simp [properFace, seedCliqueVertex, seedCliqueFace, sectorVertices, sectorEdges, edgeVertices, sectorCircleEdge, sectorLetter, seedWord, circleStart, circleEnd]
+  have hef : seedGraph.Adj seedCliqueEdge seedCliqueFace := by
+    change properFace _ _ ∨ _
+    left
+    simp [properFace, seedCliqueEdge, seedCliqueFace, sectorEdges, sectorCircleEdge, sectorLetter, seedWord]
+  exact ⟨cliqueSquare seedGraph _ _ _ hve hvf hef,
+    cliqueSquare_injective seedGraph _ _ _ hve hvf hef⟩
+
+section CupCocycle
+open groupCohomology
+variable {G : Type} [Group G]
+
+/-- The cup product of two integral exponent homomorphisms. -/
+def exponentCup (a b : G →* Multiplicative ℤ) : cocycles₂ (Rep.trivial ℤ G ℤ) :=
+  ⟨fun p => (a p.1).toAdd * (b p.2).toAdd, by
+    rw [mem_cocycles₂_iff]
+    intro g h j
+    simp only [map_mul, toAdd_mul]
+    change ((a g).toAdd + (a h).toAdd) * (b j).toAdd + (a g).toAdd * (b h).toAdd =
+      (a h).toAdd * (b j).toAdd + (a g).toAdd * ((b h).toAdd + (b j).toAdd)
+    ring⟩
+
+/-- Antisymmetrizing on a commuting pair detects a nonzero integral class. -/
+theorem exponentCup_not_coboundary (a b : G →* Multiplicative ℤ) (g h : G)
+    (hc : Commute g h)
+    (hag : (a g).toAdd = 1) (hah : (a h).toAdd = 0)
+    (hbg : (b g).toAdd = 0) (hbh : (b h).toAdd = 1) :
+    ⇑(exponentCup a b) ∉ coboundaries₂ (Rep.trivial ℤ G ℤ) := by
+  rintro ⟨f, hf⟩
+  have h₁ := congrFun hf (g, h)
+  have h₂ := congrFun hf (h, g)
+  change f h - f (g*h) + f g = (a g).toAdd * (b h).toAdd at h₁
+  change f g - f (h*g) + f h = (a h).toAdd * (b g).toAdd at h₂
+  rw [hag, hbh] at h₁
+  rw [hah, hbg, ← hc.eq] at h₂
+  change (f h : ℤ) - f (g*h) + f g = 1 * 1 at h₁
+  change (f g : ℤ) - f (g*h) + f h = 0 * 0 at h₂
+  linarith
+
+theorem exponentCup_class_ne_zero (a b : G →* Multiplicative ℤ) (g h : G)
+    (hc : Commute g h)
+    (hag : (a g).toAdd = 1) (hah : (a h).toAdd = 0)
+    (hbg : (b g).toAdd = 0) (hbh : (b h).toAdd = 1) :
+    H2π _ (exponentCup a b) ≠ 0 := by
+  intro he
+  exact exponentCup_not_coboundary a b g h hc hag hah hbg hbh
+    ((H2π_eq_zero_iff (A := Rep.trivial ℤ G ℤ) (exponentCup a b)).mp he)
+
+end CupCocycle
+
+section CliqueCohomology
+open groupCohomology
+variable {V : Type} (L : SimpleGraph V)
+
+theorem clique_exponent_class_nonzero (v w b : V)
+    (hvw : L.Adj v w) (hvb : L.Adj v b) (hwb : L.Adj w b) :
+    ∃ x : H2 (Rep.trivial ℤ (height L).ker ℤ), x ≠ 0 := by
+  classical
+  let a := (generatorExponent L v).comp (height L).ker.subtype
+  let d := (generatorExponent L w).comp (height L).ker.subtype
+  have hvw' := L.ne_of_adj hvw
+  have hvb' := L.ne_of_adj hvb
+  have hwb' := L.ne_of_adj hwb
+  refine ⟨H2π _ (exponentCup a d), exponentCup_class_ne_zero a d
+    (heightDifference L v b) (heightDifference L w b)
+    (heightDifference_commute L v w b hvw hvb hwb) ?_ ?_ ?_ ?_⟩
+  all_goals simp [a, d, heightDifference, hvw', hvw'.symm, hvb'.symm, hwb'.symm]
+
+end CliqueCohomology
+
+
+open CategoryTheory CategoryTheory.Limits
+
+section
+variable {C D : Type*} [Category* C] [Category* D] [Abelian C] [Abelian D]
+    {X Y : C} (P : ProjectiveResolution X) (F : C ⥤ D)
+    [F.Additive] [F.Full] [F.Faithful]
+    [F.PreservesProjectiveObjects] [F.PreservesHomology]
+
+/-- Vanishing in the literal projective-dimension sense forces every
+2-cocycle in a projective resolution to be a coboundary; the fully faithful
+exact functor permits us to use this for the representation/group-ring equivalence. -/
+theorem cocycle_boundary_of_projectiveDimension_lt_two
+    [HasProjectiveDimensionLT (F.obj X) 2]
+    (f : P.complex.X 2 ⟶ Y) (hf : P.complex.d 3 2 ≫ f = 0) :
+    ∃ g : P.complex.X 1 ⟶ Y, P.complex.d 2 1 ≫ g = f := by
+  let := HasExt.standard D
+  let Q := F.mapProjectiveResolution P
+  have hmf : Q.complex.d 3 2 ≫ F.map f = 0 := by
+    change F.map (P.complex.d 3 2) ≫ F.map f = 0
+    rw [← F.map_comp, hf, F.map_zero]
+  have hc := Abelian.Ext.eq_zero_of_hasProjectiveDimensionLT (Q.extMk (F.map f) 3 rfl hmf) 2 (by rfl)
+  obtain ⟨g, hg⟩ := (Q.extMk_eq_zero_iff (F.map f) 3 rfl hmf 1 rfl).mp hc
+  change F.obj (P.complex.X 1) ⟶ F.obj Y at g
+  change F.map (P.complex.d 2 1) ≫ g = F.map f at hg
+  refine ⟨F.preimage g, F.map_injective ?_⟩
+  rw [F.map_comp, F.map_preimage]
+  exact hg
+
+/-- The same vanishing expressed in the library's older linear-Yoneda model
+of Ext. This theorem proves rather than assumes the bridge between the two APIs. -/
+theorem isZero_homology_two_of_projectiveDimension_lt_two
+    (R : Type*) [Ring R] [Linear R C]
+    [HasProjectiveDimensionLT (F.obj X) 2] :
+    IsZero ((P.complex.linearYonedaObj R Y).homology 2) := by
+  rw [← HomologicalComplex.exactAt_iff_isZero_homology]
+  rw [HomologicalComplex.exactAt_iff' _ 1 2 3 (by simp) (by simp)]
+  rw [ShortComplex.moduleCat_exact_iff]
+  intro f hf
+  change P.complex.d 3 2 ≫ f = 0 at hf
+  exact cocycle_boundary_of_projectiveDimension_lt_two P F f hf
+end
+
+open CategoryTheory CategoryTheory.Limits
+
+theorem two_le_cohomologicalDimension_of_H2_nonzero (G : Type) [Group G]
+    (hx : ∃ x : groupCohomology.H2 (Rep.trivial ℤ G ℤ), x ≠ 0) :
+    2 ≤ cohomologicalDimension G := by
+  change (2 : ℕ) ≤ CategoryTheory.projectiveDimension
+    (ModuleCat.of (MonoidAlgebra ℤ G) (Representation.trivial ℤ G ℤ).asModule)
+  rw [CategoryTheory.projectiveDimension_ge_iff]
+  intro hd
+  let F := Rep.toModuleMonoidAlgebra (k := ℤ) (G := G)
+  let P := Rep.barResolution ℤ G
+  let : F.Additive := Functor.additive_of_preserves_binary_products F
+  have : HasProjectiveDimensionLT (F.obj (Rep.trivial ℤ G ℤ)) 2 := hd
+  have hz := isZero_homology_two_of_projectiveDimension_lt_two
+    (Y := Rep.trivial ℤ G ℤ) P F ℤ
+  have hz' : IsZero (groupCohomology.H2 (Rep.trivial ℤ G ℤ)) :=
+    hz.of_iso (groupCohomologyIso (Rep.trivial ℤ G ℤ) 2 P)
+  obtain ⟨x, hx⟩ := hx
+  exact hx ((ModuleCat.isZero_iff_subsingleton.mp hz').elim x 0)
+
+theorem height_kernel_two_le_cohomologicalDimension {V : Type} (L : SimpleGraph V)
+    (v w b : V) (hvw : L.Adj v w) (hvb : L.Adj v b) (hwb : L.Adj w b) :
+    2 ≤ cohomologicalDimension (height L).ker :=
+  two_le_cohomologicalDimension_of_H2_nonzero _ (clique_exponent_class_nonzero L v w b hvw hvb hwb)
+
+theorem source_two_le_cohomologicalDimension : 2 ≤ cohomologicalDimension SourceGroup := by
+  apply height_kernel_two_le_cohomologicalDimension seedGraph seedCliqueVertex seedCliqueEdge seedCliqueFace
+  · change properFace _ _ ∨ _
+    left
+    simp [properFace, seedCliqueVertex, seedCliqueEdge, edgeVertices, circleStart]
+  · change properFace _ _ ∨ _
+    left
+    simp [properFace, seedCliqueVertex, seedCliqueFace, sectorVertices, sectorEdges, edgeVertices, sectorCircleEdge, sectorLetter, seedWord, circleStart, circleEnd]
+  · change properFace _ _ ∨ _
+    left
+    simp [properFace, seedCliqueEdge, seedCliqueFace, sectorEdges, sectorCircleEdge, sectorLetter, seedWord]
+end EilenbergGanea
+
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/CubicalCover.lean

@@ -1,822 +1,822 @@
-import OAI.AlgebraicTopology.Cubical.Realization
-import OAI.AlgebraicTopology.Cubical.ArtinChains
-import OAI.Topology.EilenbergGanea.Model
-import OAI.Topology.EilenbergGanea.SeedConnection
-import Mathlib.Topology.Covering.Quotient
-import Mathlib.Topology.Covering.Deck
-import Mathlib.Topology.Homotopy.Lifting
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set Topology
-open scoped Quaternion
-
-namespace EilenbergGanea.DescendingCube.Data
-open CubeInterpolation
-variable {G A : Type*} [Group G] {f : A → G} (D : Data f)
-variable [Fintype A]
-
-omit [Fintype A] in
-theorem cellComm (c : D.Cell) : (c.directions : Set A).Pairwise
-    (fun a b => Commute (f a) (f b)) :=
-  fun _ ha _ hb he => D.commute c.top _ _ (c.subset ha) (c.subset hb) he
-
-omit [Fintype A] in
-theorem ones_eq_filter (c : D.Cell) (x : D.Coordinates c) :
-    D.ones c x = c.directions.filter (fun a => D.fullCoord c x a = 1) := by
-  ext a
-  simp only [mem_ones,Finset.mem_filter]
-  constructor
-  · rintro ⟨ha,hx⟩
-    exact ⟨ha,by simpa only [fullCoord,dite_eq_left ha] using hx⟩
-  · rintro ⟨ha,hx⟩
-    exact ⟨ha,by simpa only [fullCoord,dite_eq_left ha] using hx⟩
-
-omit [Fintype A] in
-theorem normalize_fullCoord (c : D.Cell) (x : D.Coordinates c) :
-    CubeInterpolation.normalize (D.fullCoord c x) = D.normalizedCoord c x := by
-  funext a
-  by_cases ha : a ∈ c.directions
-  · simp only [CubeInterpolation.normalize,fullCoord,normalizedCoord,dite_eq_left ha]
-  · simp [CubeInterpolation.normalize,fullCoord,normalizedCoord,ha]
-
-def interpolation (p : D.Point) (φ : G → ℝ) : ℝ :=
-  operator f (D.active p) (D.cellComm (D.pointCell p)) p.coord φ p.top
-
-theorem active_characteristic_subset (c : D.Cell) (x : D.Coordinates c) :
-    D.active (D.characteristic c x) ⊆ c.directions := by
-  intro a ha
-  by_contra h
-  apply ((D.mem_active _ _).mp ha)
-  change D.normalizedCoord c x a = 0
-  simp [normalizedCoord,h]
-
-theorem interpolation_characteristic (c : D.Cell) (x : D.Coordinates c) (φ : G → ℝ) :
-    D.interpolation (D.characteristic c x) φ =
-      operator f c.directions (D.cellComm c) (D.fullCoord c x) φ c.top := by
-  have hsub := D.active_characteristic_subset c x
-  have he := operator_subset f (D.active (D.characteristic c x)) c.directions
-    (D.cellComm c) hsub (D.normalizedCoord c x) (by
-      intro a _ hn
-      exact not_not.mp (by simpa only [mem_active,characteristic] using hn))
-  have hn := operator_normalize f c.directions (D.cellComm c) (D.fullCoord c x)
-  simp only [normalize_fullCoord,← ones_eq_filter] at hn
-  have hh := congrArg (fun T : Module.End ℝ (G → ℝ) => T φ c.top) hn
-  rw [Module.End.mul_apply,translation_apply,he] at hh
-  exact hh.symm
-
-theorem interpolation_continuous (φ : G → ℝ) :
-    Continuous (fun p : D.Point => D.interpolation p φ) := by
-  apply D.evaluation_quotient.continuous_iff.mpr
-  apply continuous_sigma
-  intro c
-  change Continuous (fun x : D.Coordinates c => D.interpolation (D.characteristic c x) φ)
-  simp only [interpolation_characteristic]
-  exact operator_continuous f c.directions _ (D.fullCoord c)
-    (fun a _ => D.fullCoord_continuous c a) φ c.top
-
-theorem interpolation_vertex_pos (p : D.Point) :
-    0 < D.interpolation p (fun a => if a = p.top then 1 else 0) := by
-  apply operator_pos
-  · intro a _; exact ⟨p.nonneg a,p.lt_one a⟩
-  · intro a; split_ifs <;> norm_num
-  · simp
-
-omit [Fintype A] in
-theorem descending_operator_delta_zero (s : Finset A) (g : G) (hs : s ⊆ D.desc g)
-    (x : A → ℝ) (b : G) (hb : D.length g ≤ D.length b) (hne : b ≠ g) :
-    operator f s (fun a ha d hd hne => D.commute g a d (hs ha) (hs hd) hne)
-      x (fun a => if a = b then 1 else 0) g = 0 := by
-  induction s using Finset.induction_on generalizing g with
-  | empty => simp [operator_empty,hne.symm]
-  | @insert a s ha ih =>
-    have hga := hs (Finset.mem_insert_self a s)
-    have hs0 : s ⊆ D.desc g := fun d hd => hs (Finset.mem_insert_of_mem hd)
-    have hs1 : s ⊆ D.desc (g * f a) := by
-      intro d hd
-      exact D.preserve g a d hga (hs0 hd) (by intro he; exact ha (he ▸ hd))
-    have hd := D.decrease g a hga
-    have hb1 : D.length (g*f a) ≤ D.length b := by omega
-    have hne1 : b ≠ g*f a := by intro he; rw [he] at hb; omega
-    have hc : (↑(insert a s) : Set A).Pairwise (fun a d => Commute (f a) (f d)) :=
-      fun _ ha _ hd hne => D.commute g _ _ (hs ha) (hs hd) hne
-    change operator f (insert a s) hc x (fun a => if a = b then 1 else 0) g = 0
-    rw [operator_insert f s a ha,Module.End.mul_apply,blend_apply]
-    rw [ih g hs0 hb hne,ih (g*f a) hs1 hb1 hne1]
-    ring
-
-theorem interpolation_vertex_zero (p : D.Point) (b : G)
-    (hb : D.length p.top ≤ D.length b) (hne : b ≠ p.top) :
-    D.interpolation p (fun a => if a = b then 1 else 0) = 0 :=
-  D.descending_operator_delta_zero (D.active p) p.top (D.pointCell p).subset p.coord b hb hne
-
-theorem interpolation_top_eq (p q : D.Point)
-    (he : ∀ φ, D.interpolation p φ = D.interpolation q φ) : p.top = q.top := by
-  have length_le (p q : D.Point) (he : ∀ φ, D.interpolation p φ = D.interpolation q φ) :
-      D.length q.top ≤ D.length p.top := by
-    by_contra hn
-    have hlt : D.length p.top < D.length q.top := lt_of_not_ge hn
-    have hne : q.top ≠ p.top := by intro eq; rw [eq] at hlt; exact lt_irrefl _ hlt
-    have hz := D.interpolation_vertex_zero p q.top hlt.le hne
-    have hp := D.interpolation_vertex_pos q
-    rw [← he,hz] at hp
-    exact lt_irrefl _ hp
-  have h₁ := length_le p q he
-  by_contra hne
-  have hz := D.interpolation_vertex_zero q p.top h₁ hne
-  have hp := D.interpolation_vertex_pos p
-  rw [he,hz] at hp
-  exact lt_irrefl _ hp
-
-variable (exponent : A → G →* Multiplicative ℝ)
-variable (hexponent : ∀ g a, a ∈ D.desc g → ∀ b, b ∈ D.desc g →
-  (exponent a (f b)).toAdd = if a = b then 1 else 0)
-
-include hexponent in
-theorem interpolation_exponent (p : D.Point) (a : A) (ha : a ∈ D.desc p.top) :
-    D.interpolation p (fun g => (exponent a g).toAdd) =
-      (exponent a p.top).toAdd + p.coord a := by
-  refine (operator_linear f (exponent a) (D.active p)
-    (D.cellComm (D.pointCell p)) p.coord p.top).trans ?_
-  congr 1
-  have he : ∀ b ∈ D.active p, (exponent a (f b)).toAdd = if a = b then 1 else 0 :=
-    fun b hb => hexponent p.top a ha b ((D.pointCell p).subset hb)
-  simp only [Finset.sum_congr rfl (fun b hb => congrArg (fun t : ℝ => p.coord b * t) (he b hb)),
-    mul_ite,mul_one,mul_zero,Finset.sum_ite_eq]
-  by_cases h : p.coord a = 0
-  · simp [h]
-  · simp [mem_active,h]
-
-include exponent hexponent in
-theorem interpolation_injective : Function.Injective
-    (fun p : D.Point => fun φ : G → ℝ => D.interpolation p φ) := by
-  intro p q he
-  have he' : ∀ φ, D.interpolation p φ = D.interpolation q φ := fun φ => congrFun he φ
-  have hb := D.interpolation_top_eq p q he'
-  apply Point.ext hb
-  funext a
-  by_cases ha : a ∈ D.desc p.top
-  · have hq : a ∈ D.desc q.top := by simpa only [hb] using ha
-    have hh := he' (fun g => (exponent a g).toAdd)
-    rw [D.interpolation_exponent exponent hexponent p a ha,
-      D.interpolation_exponent exponent hexponent q a hq,hb] at hh
-    exact add_left_cancel hh
-  · have hp : p.coord a = 0 := by_contra (fun hn => ha (p.support a hn))
-    have hq : q.coord a = 0 := by
-      by_contra hn
-      exact ha (by simpa only [hb] using q.support a hn)
-    rw [hp,hq]
-
-include exponent hexponent in
-theorem point_t2 : T2Space D.Point :=
-  T2Space.of_injective_continuous
-    (D.interpolation_injective exponent hexponent)
-    (continuous_pi (fun φ => D.interpolation_continuous φ))
-
-end EilenbergGanea.DescendingCube.Data
-
-open Classical Topology
-namespace WeakRealization
-
-/-- A homeomorphism is induced by point-separating coordinates and actual
-continuous lifts on each characteristic disk, in both directions. -/
-def homeomorph {ι κ X Y Z : Type*} {E : ι → Type*} {F : κ → Type*}
-    [TopologicalSpace X] [TopologicalSpace Y]
-    [∀ i, TopologicalSpace (E i)] [∀ j, TopologicalSpace (F j)]
-    (e : ((i : ι) × E i) → X) (f : ((j : κ) × F j) → Y)
-    (he : IsQuotientMap e) (hf : IsQuotientMap f)
-    (J : X → Z) (K : Y → Z) (hJ : Function.Injective J) (hK : Function.Injective K)
-    (hxy : ∀ i, ∃ φ : C(E i,Y), ∀ t, K (φ t) = J (e ⟨i,t⟩))
-    (hyx : ∀ j, ∃ φ : C(F j,X), ∀ t, J (φ t) = K (f ⟨j,t⟩)) : X ≃ₜ Y := by
-  have hx : ∀ x, ∃ y, K y = J x := by
-    intro x
-    obtain ⟨⟨i,t⟩,rfl⟩ := he.surjective x
-    obtain ⟨φ,hφ⟩ := hxy i
-    exact ⟨φ t,hφ t⟩
-  have hy : ∀ y, ∃ x, J x = K y := by
-    intro y
-    obtain ⟨⟨j,t⟩,rfl⟩ := hf.surjective y
-    obtain ⟨φ,hφ⟩ := hyx j
-    exact ⟨φ t,hφ t⟩
-  let A : X → Y := fun x => (hx x).choose
-  let B : Y → X := fun y => (hy y).choose
-  have hA : ∀ x, K (A x) = J x := fun x => (hx x).choose_spec
-  have hB : ∀ y, J (B y) = K y := fun y => (hy y).choose_spec
-  refine {
-    toEquiv := {
-      toFun := A
-      invFun := B
-      left_inv := fun x => hJ ((hB (A x)).trans (hA x))
-      right_inv := fun y => hK ((hA (B y)).trans (hB y)) }
-    continuous_toFun := ?_
-    continuous_invFun := ?_ }
-  · apply he.continuous_iff.mpr
-    apply continuous_sigma
-    intro i
-    obtain ⟨φ,hφ⟩ := hxy i
-    have hh : (fun t => A (e ⟨i,t⟩)) = φ := by
-      funext t
-      exact hK ((hA _).trans (hφ t).symm)
-    change Continuous (fun t => A (e ⟨i,t⟩))
-    rw [hh]
-    exact φ.continuous
-  · apply hf.continuous_iff.mpr
-    apply continuous_sigma
-    intro j
-    obtain ⟨φ,hφ⟩ := hyx j
-    have hh : (fun t => B (f ⟨j,t⟩)) = φ := by
-      funext t
-      exact hJ ((hB _).trans (hφ t).symm)
-    change Continuous (fun t => B (f ⟨j,t⟩))
-    rw [hh]
-    exact φ.continuous
-
-end WeakRealization
-
-namespace EilenbergGanea.CubeInterpolation
-open CubicalChains.Reorientation
-variable {G V : Type*} [Group G]
-
-/-- Interpolation in a listed cube, independent of any topological model. -/
-def listed (f : V → G) (w : List V) (x : V → ℝ) : Module.End ℝ (G → ℝ) :=
-  (w.map (fun v => blend (f v) (x v))).prod
-
-@[simp] theorem listed_nil (f : V → G) (x : V → ℝ) : listed f [] x = 1 := rfl
-@[simp] theorem listed_cons (f : V → G) (v : V) (w : List V) (x : V → ℝ) :
-    listed f (v::w) x = blend (f v) (x v) * listed f w x := rfl
-
-theorem operator_listed [DecidableEq V] (f : V → G) (w : List V) (hw : w.Nodup)
-    (hc : (w.toFinset : Set V).Pairwise (fun v u => Commute (f v) (f u))) (x : V → ℝ) :
-    operator f w.toFinset hc x = listed f w x :=
-  Finset.noncommProd_toFinset w _ _ hw
-
-/-- Literal multilinear invariance under reversing selected cube directions. -/
-theorem listed_reorientation (f : V → G) (w : List (Signed V))
-    (hw : w.Pairwise (fun a b => Commute (f a.1) (f b.1)))
-    (x : Signed V → ℝ) (z : V → ℝ)
-    (hx : ∀ a ∈ w, x a = if a.2 then 1-z a.1 else z a.1) (φ : G → ℝ) (g : G) :
-    listed (step f) w x φ g = listed f (w.map Prod.fst) z φ (origin f g w) := by
-  induction w generalizing g with
-  | nil => rfl
-  | cons a w ih =>
-    obtain ⟨ha,hw⟩ := List.pairwise_cons.mp hw
-    have hxw : ∀ b ∈ w, x b = if b.2 then 1-z b.1 else z b.1 :=
-      fun b hb => hx b (by simp [hb])
-    simp only [listed_cons,List.map_cons,Module.End.mul_apply,blend_apply,
-      ih hw hxw,hx a (by simp),step,origin]
-    cases hb : a.2
-    · simp only [Bool.false_eq_true,↓reduceIte]
-      rw [origin_shift f g (f a.1) w ha]
-    · simp only [↓reduceIte]
-      rw [origin_shift f g (f a.1)⁻¹ w (fun b hb => (ha b hb).inv_left),
-        inv_mul_cancel_right]
-      ring
-
-end EilenbergGanea.CubeInterpolation
-
-namespace EilenbergGanea.ArtinCubicalTopology
-open TraceWords CubeInterpolation CubicalChains.Reorientation
-variable {V : Type*} [Fintype V] (L : SimpleGraph V)
-
-local instance : Std.Symm L.Adj := ⟨fun _ _ h => h.symm⟩
-local instance : Std.Irrefl L.Adj := ⟨fun _ => L.irrefl⟩
-
-abbrev f := artinGenerator L
-abbrev comm := fun v w h => adjacent_generators_commute L (v := v) (w := w) h
-abbrev D := artinDescendingData L
-abbrev Positive := CubeRealization.Point L (f L) (comm L)
-abbrev Descending := (D L).Point
-
-omit [Fintype V] in
-@[simp] theorem step_eq (a : Letter V) : step (f L) a = letterValue (f L) (invLetter a) := by
-  cases a with | mk v b => cases b <;> rfl
-
-def exponent (v : V) : ArtinGroup L →* Multiplicative ℝ :=
-  artinLift L (fun w => Multiplicative.ofAdd (if v = w then (1 : ℝ) else 0))
-    (fun _ _ _ => Commute.all _ _)
-
-omit [Fintype V] in
-@[simp] theorem exponent_generator (v w : V) :
-    (exponent L v (f L w)).toAdd = if v = w then 1 else 0 := by
-  simp [exponent,f]
-
-omit [Fintype V] in
-@[simp] theorem height_generator (v : V) : (height L (f L v)).toAdd = 1 := by
-  exact congrArg Multiplicative.toAdd (PresentedGroup.toGroup.of _)
-
-def signedExponent (a : Letter V) : ArtinGroup L →* Multiplicative ℝ :=
-  artinLift L (fun v => Multiplicative.ofAdd
-    (if a.1 = v then (if a.2 then (-1 : ℝ) else 1) else 0))
-    (fun _ _ _ => Commute.all _ _)
-
-theorem signedExponent_step (g : ArtinGroup L) (a : Letter V) (ha : a ∈ (D L).desc g)
-    (b : Letter V) (hb : b ∈ (D L).desc g) :
-    (signedExponent L a (letterValue (f L) (invLetter b))).toAdd = if a = b then 1 else 0 := by
-  by_cases he : a = b
-  · subst b
-    rcases a with ⟨v,b⟩
-    cases b <;> simp [signedExponent,letterValue,invLetter,f]
-  · have hi : Independent L a b := rightDescents_pairwise L g
-      (by simpa [D,artinDescendingData] using ha) (by simpa [D,artinDescendingData] using hb) he
-    have hv : a.1 ≠ b.1 := hi.ne
-    rcases b with ⟨v,b⟩
-    cases b <;> simp [signedExponent,letterValue,invLetter,f,hv,he]
-
-theorem positive_interp_inj : Function.Injective
-    (fun p : Positive L => fun φ => CubeRealization.interpolation L (f L) (comm L) p φ) :=
-  CubeRealization.interpolation_injective L (f L) (comm L) (height L) (height_generator L)
-    (exponent L) (exponent_generator L)
-
-theorem descending_interp_inj : Function.Injective
-    (fun p : Descending L => fun φ => (D L).interpolation p φ) :=
-  (D L).interpolation_injective (signedExponent L) (by
-    intro g a ha b hb
-    by_cases he : a = b <;> simpa [he] using signedExponent_step L g a ha b hb)
-
-def flip (b : Bool) (t : unitInterval) : unitInterval :=
-  if b then unitInterval.symm t else t
-
-@[simp] theorem flip_flip (b : Bool) (t : unitInterval) : flip b (flip b t) = t := by
-  cases b <;> simp [flip]
-
-theorem flip_continuous (b : Bool) : Continuous (flip b) := by
-  cases b
-  · exact continuous_id
-  · exact unitInterval.continuous_symm
-
-def vertexEquiv (w : Word V) (hw : (w.map Prod.fst).Nodup) :
-    w.toFinset ≃ (w.map Prod.fst).toFinset :=
-  Equiv.ofBijective (fun a => ⟨a.val.1,by
-    exact List.mem_toFinset.mpr (List.mem_map.mpr ⟨a.val,List.mem_toFinset.mp a.property,rfl⟩)⟩) (by
-    constructor
-    · intro a b he
-      apply Subtype.ext
-      exact (List.nodup_map_iff_inj_on (hw.of_map Prod.fst)).mp hw
-        a.val (List.mem_toFinset.mp a.property) b.val (List.mem_toFinset.mp b.property) (congrArg Subtype.val he)
-    · intro v
-      obtain ⟨a,ha,he⟩ := List.mem_map.mp (List.mem_toFinset.mp v.property)
-      exact ⟨⟨a,by simpa using ha⟩,Subtype.ext he⟩)
-
-omit [Fintype V] in
-@[simp] theorem vertexEquiv_val (w : Word V) (hw : (w.map Prod.fst).Nodup)
-    (a : w.toFinset) : (vertexEquiv w hw a).val = a.val.1 := rfl
-
-def toPositiveCoords (w : Word V) (hw : (w.map Prod.fst).Nodup)
-    (x : w.toFinset → unitInterval) : (w.map Prod.fst).toFinset → unitInterval :=
-  fun v => flip ((vertexEquiv w hw).symm v).val.2 (x ((vertexEquiv w hw).symm v))
-
-omit [Fintype V] in
-@[simp] theorem toPositiveCoords_apply (w : Word V) (hw : (w.map Prod.fst).Nodup)
-    (x : w.toFinset → unitInterval) (a : w.toFinset) :
-    toPositiveCoords w hw x (vertexEquiv w hw a) = flip a.val.2 (x a) := by
-  simp [toPositiveCoords]
-
-omit [Fintype V] in
-theorem toPositiveCoords_continuous (w : Word V) (hw : (w.map Prod.fst).Nodup) :
-    Continuous (toPositiveCoords w hw) :=
-  continuous_pi (fun _ => (flip_continuous _).comp (continuous_apply _))
-
-def toDescendingCoords (w : Word V) (hw : (w.map Prod.fst).Nodup)
-    (z : (w.map Prod.fst).toFinset → unitInterval) : w.toFinset → unitInterval :=
-  fun a => flip a.val.2 (z (vertexEquiv w hw a))
-
-omit [Fintype V] in
-theorem toDescendingCoords_continuous (w : Word V) (hw : (w.map Prod.fst).Nodup) :
-    Continuous (toDescendingCoords w hw) :=
-  continuous_pi (fun _ => (flip_continuous _).comp (continuous_apply _))
-
-omit [Fintype V] in
-@[simp] theorem toPositive_toDescending (w : Word V) (hw : (w.map Prod.fst).Nodup)
-    (z : (w.map Prod.fst).toFinset → unitInterval) :
-    toPositiveCoords w hw (toDescendingCoords w hw z) = z := by
-  funext v
-  obtain ⟨a,rfl⟩ := (vertexEquiv w hw).surjective v
-  simp [toDescendingCoords]
-
-def positiveCell (g : ArtinGroup L) (w : Word V) (hw : (w.map Prod.fst).Pairwise L.Adj) :
-    CubeRealization.Cell (G := ArtinGroup L) L where
-  base := origin (f L) g w
-  directions := (w.map Prod.fst).toFinset
-  clique := fun _ ha _ hb he => hw.forall (List.mem_toFinset.mp ha) (List.mem_toFinset.mp hb) he
-
-def descendingCell (g : ArtinGroup L) (w : Word V)
-    (hd : ∀ a ∈ w, a ∈ rightDescents L g) : (D L).Cell where
-  top := g
-  directions := w.toFinset
-  subset a ha := by simpa [D,artinDescendingData] using hd a (List.mem_toFinset.mp ha)
-
-theorem interpolation_reorientation (g : ArtinGroup L) (w : Word V)
-    (hw : (w.map Prod.fst).Pairwise L.Adj) (hd : ∀ a ∈ w, a ∈ rightDescents L g)
-    (x : (D L).Coordinates (descendingCell L g w hd)) (φ : ArtinGroup L → ℝ) :
-    (D L).interpolation ((D L).characteristic (descendingCell L g w hd) x) φ =
-      CubeRealization.interpolation L (f L) (comm L)
-        (CubeRealization.characteristic (f L) (comm L) (positiveCell L g w hw)
-          (toPositiveCoords w hw.nodup x)) φ := by
-  refine ((D L).interpolation_characteristic (descendingCell L g w hd) x φ).trans ?_
-  refine Eq.trans ?_ (CubeRealization.interpolation_characteristic L (f L) (comm L)
-    (positiveCell L g w hw) (toPositiveCoords w hw.nodup x) φ).symm
-  have hf : (fun a : Letter V => letterValue (artinGenerator L) (invLetter a)) = step (f L) :=
-    funext (fun a => (step_eq L a).symm)
-  erw [operator_listed _ w (hw.nodup.of_map Prod.fst),operator_listed _ _ hw.nodup]
-  have hop := congrArg (fun k : Letter V → ArtinGroup L => listed k w) hf
-  rw [hop]
-  apply listed_reorientation
-  · exact (List.pairwise_map.mp hw).imp (fun hh => adjacent_generators_commute L hh)
-  · intro a ha
-    let aa : w.toFinset := ⟨a,by simpa using ha⟩
-    have h₁ : (D L).fullCoord (descendingCell L g w hd) x a = (x aa : ℝ) := by
-      exact (D L).fullCoord_mem (descendingCell L g w hd) x aa
-    have h₂ : CubeRealization.fullCoord (positiveCell L g w hw)
-        (toPositiveCoords w hw.nodup x) a.1 = (flip a.2 (x aa) : ℝ) := by
-      have hh := CubeRealization.fullCoord_mem (positiveCell L g w hw)
-        (toPositiveCoords w hw.nodup x) (vertexEquiv w hw.nodup aa)
-      exact hh.trans (congrArg (fun z : unitInterval => (z : ℝ))
-        (toPositiveCoords_apply w hw.nodup x aa))
-    rw [h₁,h₂]
-    cases h : a.2 <;> simp [flip]
-
-theorem descendingCell_rep (c : (D L).Cell) :
-    ∃ (g : ArtinGroup L) (w : Word V) (_hw : (w.map Prod.fst).Pairwise L.Adj)
-      (hd : ∀ a ∈ w, a ∈ rightDescents L g), descendingCell L g w hd = c := by
-  let w := c.directions.toList
-  have hd : ∀ a ∈ w, a ∈ rightDescents L c.top := by
-    intro a ha
-    have hh := c.subset (Finset.mem_toList.mp ha)
-    simpa [D,artinDescendingData] using hh
-  have hw : (w.map Prod.fst).Pairwise L.Adj := by
-    apply List.pairwise_map.mpr
-    apply c.directions.nodup_toList.pairwise_of_forall_ne
-    intro a ha b hb he
-    exact rightDescents_pairwise L c.top (hd a ha) (hd b hb) he
-  refine ⟨c.top,w,hw,hd,?_⟩
-  cases c with
-  | mk g s hs =>
-    dsimp only [descendingCell,w]
-    congr
-    exact s.toList_toFinset
-
-omit [Fintype V] in
-theorem positiveCell_rep (c : CubeRealization.Cell (G := ArtinGroup L) L) :
-    ∃ (g : ArtinGroup L) (w : Word V) (hw : (w.map Prod.fst).Pairwise L.Adj)
-      (_hd : ∀ a ∈ w, a ∈ rightDescents L g), positiveCell L g w hw = c := by
-  let v := c.directions.toList
-  have hv : v.Pairwise L.Adj := c.directions.nodup_toList.pairwise_of_forall_ne
-    (fun a ha b hb he => c.clique (Finset.mem_toList.mp ha) (Finset.mem_toList.mp hb) he)
-  let o := CubeDescents.orient L c.base v
-  have hw : (o.2.map Prod.fst).Pairwise L.Adj := by
-    simpa only [o,CubeDescents.orient_map] using hv
-  have hd : ∀ a ∈ o.2, a ∈ rightDescents L o.1 := CubeDescents.orient_descents L c.base v hv
-  refine ⟨o.1,o.2,hw,hd,?_⟩
-  have hb : origin (f L) o.1 o.2 = c.base := CubeDescents.origin_orient L c.base v hv
-  have hs : (o.2.map Prod.fst).toFinset = c.directions := by
-    simp only [o,CubeDescents.orient_map,v,Finset.toList_toFinset]
-  cases c with
-  | mk g s hc =>
-    change CubeRealization.Cell.mk (origin (f L) o.1 o.2) (o.2.map Prod.fst).toFinset _ = _
-    congr
-
-theorem descending_local_lift (c : (D L).Cell) :
-    ∃ φ : C((D L).Coordinates c,Positive L), ∀ x,
-      (fun ψ => CubeRealization.interpolation L (f L) (comm L) (φ x) ψ) =
-        (fun ψ => (D L).interpolation ((D L).characteristic c x) ψ) := by
-  obtain ⟨g,w,hw,hd,rfl⟩ := descendingCell_rep L c
-  refine ⟨⟨fun x => CubeRealization.characteristic (f L) (comm L) (positiveCell L g w hw)
-    (toPositiveCoords w hw.nodup x),
-    (CubeRealization.characteristic_continuous (f L) (comm L) (positiveCell L g w hw)).comp
-      (toPositiveCoords_continuous w hw.nodup)⟩,?_⟩
-  intro x
-  funext ψ
-  exact (interpolation_reorientation L g w hw hd x ψ).symm
-
-theorem positive_local_lift (c : CubeRealization.Cell (G := ArtinGroup L) L) :
-    ∃ φ : C(CubeRealization.Coordinates c,Descending L), ∀ x,
-      (fun ψ => (D L).interpolation (φ x) ψ) =
-        (fun ψ => CubeRealization.interpolation L (f L) (comm L)
-          (CubeRealization.characteristic (f L) (comm L) c x) ψ) := by
-  obtain ⟨g,w,hw,hd,rfl⟩ := positiveCell_rep L c
-  refine ⟨⟨fun x => (D L).characteristic (descendingCell L g w hd) (toDescendingCoords w hw.nodup x),
-    ((D L).characteristic_continuous (descendingCell L g w hd)).comp
-      (toDescendingCoords_continuous w hw.nodup)⟩,?_⟩
-  intro x
-  funext ψ
-  change (D L).interpolation ((D L).characteristic (descendingCell L g w hd)
-    (toDescendingCoords w hw.nodup x)) ψ = _
-  change (w.map Prod.fst).toFinset → unitInterval at x
-  simpa only [toPositive_toDescending] using interpolation_reorientation L g w hw hd
-    (toDescendingCoords w hw.nodup x) ψ
-
-/-- Reorientation preserves the actual weak topologies, not only cellular chains. -/
-def reorient : Descending L ≃ₜ Positive L :=
-  WeakRealization.homeomorph (D L).evaluation (CubeRealization.evaluation (f L) (comm L))
-    (D L).evaluation_quotient (CubeRealization.evaluation_quotient (f L) (comm L))
-    (fun p ψ => (D L).interpolation p ψ)
-    (fun p ψ => CubeRealization.interpolation L (f L) (comm L) p ψ)
-    (descending_interp_inj L) (positive_interp_inj L)
-    (descending_local_lift L) (positive_local_lift L)
-
-variable [Nonempty V]
-
-/-- The ordinary positive-cube CW realization is genuinely contractible. -/
-theorem positive_contractible : ContractibleSpace (Positive L) := by
-  let : ContractibleSpace (Descending L) := artinDescending_contractible L
-  exact (reorient L).symm.contractibleSpace
-
-end EilenbergGanea.ArtinCubicalTopology
-
-open Classical Set Topology
-namespace CompactChart
-
-/-- A characteristic map which has singleton fibers over its open cell has a
-continuous inverse there. Compactness is used only on the closed disk; no
-regularity of the attaching sphere is required. -/
-theorem continuousOn_symm {A X : Type*} [TopologicalSpace A] [TopologicalSpace X]
-    [T2Space X] (e : PartialEquiv A X) (K : Set A) (hK : IsCompact K)
-    (hsub : e.source ⊆ K) (hc : ContinuousOn e K)
-    (hboundary : ∀ a ∈ K, e a ∈ e.target → a ∈ e.source) :
-    ContinuousOn e.symm e.target := by
-  let : CompactSpace K := isCompact_iff_compactSpace.mp hK
-  let f : K → X := fun a => e a
-  have hf : Continuous f := continuousOn_iff_continuous_domRestrict.mp hc
-  let r := e.target.restrictPreimage f
-  have hr : Function.Surjective r := by
-    intro x
-    refine ⟨⟨⟨e.symm x,hsub (e.map_target x.property)��,?_⟩,?_⟩
-    · change e (e.symm x) ∈ e.target
-      rw [e.right_inv x.property]
-      exact x.property
-    · exact Subtype.ext (e.right_inv x.property)
-  have hq : IsQuotientMap r :=
-    (hf.isClosedMap.restrictPreimage e.target).isQuotientMap hf.restrictPreimage hr
-  apply continuousOn_iff_continuous_domRestrict.mpr
-  apply hq.continuous_iff.mpr
-  have he : (fun x : e.target => e.symm x) ∘ r =
-      (fun a : f ⁻¹' e.target => (a.val.val : A)) := by
-    funext a
-    exact e.left_inv (hboundary a.val a.val.property a.property)
-  change Continuous ((fun x : e.target => e.symm x) ∘ r)
-  rw [he]
-  exact continuous_subtype_val.comp continuous_subtype_val
-
-end CompactChart
-
-namespace EilenbergGanea.CubeRealization
-
-section
-open Metric
-variable {G V : Type*} [Group G] [Fintype V]
-variable {L : SimpleGraph V} (f : V → G)
-variable (comm : ∀ v w, L.Adj v w → Commute (f v) (f w))
-
-theorem chartInverse_continuousOn_of_t2 [T2Space (Point (G := G) L f comm)]
-    {n : ℕ} (c : DimCell (G := G) (L := L) n) :
-    ContinuousOn (chart f comm c).symm (chart f comm c).target := by
-  apply CompactChart.continuousOn_symm (chart f comm c) (closedBall 0 1)
-    (isCompact_closedBall _ _) ball_subset_closedBall (chart_continuous f comm c).continuousOn
-  intro t ht he
-  change pointCell (chart f comm c t) = c.val at he
-  change t ∈ ball 0 1
-  by_contra hn
-  have hs : t ∈ sphere 0 1 := mem_sphere.mpr
-    (le_antisymm (mem_closedBall.mp ht) (le_of_not_gt hn))
-  have hd := boundary_dimension_lt f comm c hs
-  rw [he,c.property] at hd
-  exact (lt_irrefl _) hd
-
-/-- The ordinary CW structure follows from the genuine normalized-cube gluing;
-only Hausdorffness and compact characteristic disks are used. -/
-@[instance_reducible] def cwComplexOfT2 [T2Space (Point (G := G) L f comm)] : CWComplex (univ : Set (Point (G := G) L f comm)) where
-  cell := DimCell (G := G) (L := L)
-  map := fun _ c => chart f comm c
-  source_eq := fun _ _ => rfl
-  continuousOn := fun _ c => (chart_continuous f comm c).continuousOn
-  continuousOn_symm := fun _ c => chartInverse_continuousOn_of_t2 f comm c
-  pairwiseDisjoint' := by
-    intro a _ b _ hab
-    change Disjoint (chart f comm a.2 '' ball 0 1) (chart f comm b.2 '' ball 0 1)
-    rw [chart_image_ball,chart_image_ball]
-    apply Set.disjoint_left.mpr
-    intro p ha hb
-    apply hab
-    have he : a.2.val = b.2.val := ha.symm.trans hb
-    have hn : a.1 = b.1 := a.2.property.symm.trans ((congrArg (fun c => c.directions.card) he).trans b.2.property)
-    cases a with | mk n a =>
-      cases b with | mk m b =>
-        dsimp at hn
-        subst m
-        exact congrArg (Sigma.mk n) (Subtype.ext he)
-  mapsTo' := chart_mapsTo f comm
-  closed' := by
-    intro A _ hA
-    apply (evaluation_quotient f comm).isClosed_preimage.mp
-    rw [isClosed_sigma_iff]
-    intro c
-    let d : DimCell (G := G) (L := L) c.directions.card := ⟨c,rfl⟩
-    have hh := (hA _ d).preimage (characteristic_continuous f comm c)
-    rw [chart_image_disk] at hh
-    change IsClosed (characteristic f comm c ⁻¹' (A ∩ Set.range (characteristic f comm c))) at hh
-    change IsClosed (characteristic f comm c ⁻¹' A)
-    simpa only [Set.preimage_inter,Set.preimage_range,Set.inter_univ] using hh
-  union' := by
-    apply Set.eq_univ_of_forall
-    intro p
-    let d : DimCell (G := G) (L := L) (pointCell p).directions.card := ⟨pointCell p,rfl⟩
-    apply Set.mem_iUnion.mpr
-    refine ⟨_,Set.mem_iUnion.mpr ⟨d,?_⟩⟩
-    rw [chart_image_disk]
-    exact ⟨pointCoordinates p,characteristic_point f comm p⟩
-
-end
-
-variable {G Q V : Type*} [Group G] [Group Q] [Fintype V]
-variable {L : SimpleGraph V} (f : V → G)
-variable (comm : ∀ v w, L.Adj v w → Commute (f v) (f w))
-variable (q : G →* Q)
-
-include comm in
-omit [Fintype V] in
-theorem mappedComm : ∀ v w, L.Adj v w → Commute ((q ∘ f) v) ((q ∘ f) w) :=
-  fun v w hvw => (comm v w hvw).map q
-
-/-- Applying a group homomorphism only to the initial vertices of cubes. -/
-def mapPoint (p : Point (G := G) L f comm) : Point (G := Q) L (q ∘ f) (mappedComm f comm q) where
-  base := q p.base
-  coord := p.coord
-  nonneg := p.nonneg
-  lt_one := p.lt_one
-  clique := p.clique
-
-def mapCell (c : Cell (G := G) L) : Cell (G := Q) L where
-  base := q c.base
-  directions := c.directions
-  clique := c.clique
-
-omit [Fintype V] in
-@[simp] theorem mapPoint_base (p : Point (G := G) L f comm) :
-    (mapPoint f comm q p).base = q p.base := rfl
-
-omit [Fintype V] in
-@[simp] theorem mapPoint_coord (p : Point (G := G) L f comm) :
-    (mapPoint f comm q p).coord = p.coord := rfl
-
-omit [Fintype V] in
-@[simp] theorem mapPoint_characteristic (c : Cell (G := G) L) (x : Coordinates c) :
-    mapPoint f comm q (characteristic f comm c x) =
-      characteristic (q ∘ f) (mappedComm f comm q) (mapCell q c) x := by
-  apply Point.ext
-  · change q (c.base * product f comm (ones c x) _) =
-      q c.base * product (q ∘ f) (mappedComm f comm q) (ones c x) _
-    rw [map_mul]
-    congr 1
-    exact Finset.map_noncommProd _ _ _ q
-  · rfl
-
-theorem mapPoint_continuous : Continuous (mapPoint f comm q) := by
-  apply (evaluation_quotient f comm).continuous_iff.mpr
-  apply continuous_sigma
-  intro c
-  change Continuous (fun x => mapPoint f comm q (characteristic f comm c x))
-  simp only [mapPoint_characteristic]
-  exact characteristic_continuous (q ∘ f) (mappedComm f comm q) (mapCell q c)
-
-omit [Fintype V] in
-theorem mapCell_surjective (hq : Function.Surjective q) : Function.Surjective (mapCell (L := L) q) := by
-  rintro ⟨a,s,hs⟩
-  obtain ⟨g,rfl⟩ := hq a
-  exact ⟨⟨g,s,hs⟩,rfl⟩
-
-omit [Fintype V] in
-theorem mapPoint_surjective (hq : Function.Surjective q) : Function.Surjective (mapPoint f comm q) := by
-  rintro ⟨a,x,hx,ht,hc⟩
-  obtain ⟨g,rfl⟩ := hq a
-  exact ⟨⟨g,x,hx,ht,hc⟩,rfl⟩
-
-/-- The base-label quotient is a genuine topological quotient for the weak
-cubical topologies, proved by lifting each actual characteristic disk. -/
-theorem mapPoint_quotient (hq : Function.Surjective q) : IsQuotientMap (mapPoint f comm q) := by
-  apply isQuotientMap_iff_isClosed.mpr
-  refine ⟨mapPoint_surjective f comm q hq,?_⟩
-  intro A
-  constructor
-  · exact fun hA => hA.preimage (mapPoint_continuous f comm q)
-  · intro hA
-    apply (evaluation_quotient (q ∘ f) (mappedComm f comm q)).isClosed_preimage.mp
-    rw [isClosed_sigma_iff]
-    intro c
-    obtain ⟨d,rfl⟩ := mapCell_surjective q hq c
-    have hh := hA.preimage (characteristic_continuous f comm d)
-    change IsClosed ((fun x => mapPoint f comm q (characteristic f comm d x)) ⁻¹' A) at hh
-    change IsClosed ((fun x : Coordinates d =>
-      characteristic (q ∘ f) (mappedComm f comm q) (mapCell q d) x) ⁻¹' A)
-    simpa only [mapPoint_characteristic] using hh
-
-omit [Fintype V] in
-/-- The literal group-kernel orbits are exactly the fibers of the cube map. -/
-theorem mapPoint_eq_iff_orbit (p r : Point (G := G) L f comm) :
-    mapPoint f comm q p = mapPoint f comm q r ↔ p ∈ MulAction.orbit q.ker r := by
-  constructor
-  · intro he
-    have hb : q p.base = q r.base := congrArg Point.base he
-    have hx : p.coord = r.coord := congrArg (fun z : Point (G := Q) L (q ∘ f)
-      (mappedComm f comm q) => z.coord) he
-    let g : q.ker := ⟨p.base * r.base⁻¹,by simp only [MonoidHom.mem_ker,map_mul,map_inv,hb,mul_inv_cancel]⟩
-    refine ⟨g,?_⟩
-    apply Point.ext
-    · change (p.base * r.base⁻¹) * r.base = p.base
-      simp [mul_assoc]
-    · exact hx.symm
-  · rintro ⟨g,rfl⟩
-    apply Point.ext
-    · change q ((g : G) * r.base) = q r.base
-      rw [map_mul,g.property,one_mul]
-    · rfl
-
-/-- The quotient homeomorphism respects the actual weak topologies. -/
-def kernelQuotientHomeomorph (hq : Function.Surjective q) :
-    Quotient (MulAction.orbitRel q.ker (Point (G := G) L f comm)) ≃ₜ
-      Point (G := Q) L (q ∘ f) (mappedComm f comm q) :=
-  (Homeomorph.Quotient.congrRight (fun p r => (mapPoint_eq_iff_orbit f comm q p r).symm)).trans
-    (Topology.IsQuotientMap.homeomorph (f := ⟨mapPoint f comm q,mapPoint_continuous f comm q⟩)
-      (mapPoint_quotient f comm q hq))
-
-variable (h : G →* Multiplicative ℤ) (hf : ∀ v, (h (f v)).toAdd = 1)
-variable (exponent : V → G →* Multiplicative ℝ)
-variable (hexponent : ∀ v w, (exponent v (f w)).toAdd = if v = w then 1 else 0)
-
-include h hf exponent hexponent in
-theorem quotientPoint_t2 (hq : Function.Surjective q) :
-    T2Space (Point (G := Q) L (q ∘ f) (mappedComm f comm q)) := by
-  let := point_t2 L f comm h hf exponent hexponent
-  let := point_weaklyLocallyCompact L f comm
-  exact (kernelQuotientHomeomorph f comm q hq).t2Space
-
-include h hf exponent hexponent in
-theorem mapPoint_covering (hq : Function.Surjective q) :
-    IsQuotientCoveringMap (mapPoint f comm q) q.ker where
-  toIsQuotientMap := mapPoint_quotient f comm q hq
-  continuous_const_smul := continuous_const_smul
-  apply_eq_iff_mem_orbit := mapPoint_eq_iff_orbit f comm q _ _
-  disjoint := (subgroup_quotient_covering f comm h hf exponent hexponent q.ker).disjoint
-
-end EilenbergGanea.CubeRealization
-
-namespace EilenbergGanea
-universe u
-namespace ArtinCubicalTopology
-open CubeRealization
-variable {V : Type u} [Fintype V] [Nonempty V] (L : SimpleGraph V)
-
-omit [Fintype V] in
-theorem height_surjective : Function.Surjective (height L) := by
-  intro n
-  let v : V := Classical.choice inferInstance
-  refine ⟨f L v ^ n.toAdd,?_⟩
-  rw [map_zpow]
-  apply Multiplicative.toAdd.injective
-  change n.toAdd • (height L (f L v)).toAdd = n.toAdd
-  simp only [height_generator,zsmul_eq_mul,mul_one,Int.cast_id]
-
-def HeightQuotient := Point L ((height L) ∘ f L) (mappedComm (f L) (comm L) (height L))
-
-instance : TopologicalSpace (HeightQuotient L) := inferInstanceAs (TopologicalSpace
-  (Point L ((height L) ∘ f L) (mappedComm (f L) (comm L) (height L))))
-
-/-- An ordinary cubical classifying space for the height kernel, with one cube
-for every integral level and clique. Contractibility above is the explicit
-word-length contraction, and the fundamental-group identification reuses
-Mathlib's checked covering-monodromy equivalence. -/
-theorem height_hasClassifyingSpace (d : ℕ)
-    (hd : ∀ s : Finset V, (s : Set V).Pairwise L.Adj → s.card ≤ d) :
-    HasClassifyingSpace.{u} (height L).ker d := by
-  let X := Point L ((height L) ∘ f L) (mappedComm (f L) (comm L) (height L))
-  let E := Positive L
-  let : ContractibleSpace E := positive_contractible L
-  let : T2Space X := quotientPoint_t2 (f L) (comm L) (height L)
-    (height L) (height_generator L) (exponent L) (exponent_generator L) (height_surjective L)
-  let : Topology.CWComplex (Set.univ : Set X) :=
-    cwComplexOfT2 ((height L) ∘ f L) (mappedComm (f L) (comm L) (height L))
-  let p : E → X := mapPoint (f L) (comm L) (height L)
-  have hp : IsQuotientCoveringMap p (height L).ker :=
-    mapPoint_covering (f L) (comm L) (height L) (height L) (height_generator L)
-      (exponent L) (exponent_generator L) (height_surjective L)
-  have hs : Function.Surjective p := mapPoint_surjective (f L) (comm L) (height L) (height_surjective L)
-  let : PathConnectedSpace X := hs.pathConnectedSpace hp.continuous
-  let e : E := Classical.choice (inferInstance : Nonempty E)
-  refine ⟨X,inferInstance,inferInstance,inferInstance,?_,inferInstance,p e,?_,
-    E,inferInstance,inferInstance,p,hp.isCoveringMap,hs⟩
-  · intro k hk
-    exact ⟨fun c => Nat.not_lt_of_ge ((c.property.symm ▸ hd c.val.directions c.val.clique)) hk⟩
-  · exact ⟨(hp.fundamentalGroupEquiv ⟨e,rfl⟩).trans (MulEquiv.inv' ((height L).ker)).symm⟩
-
-end ArtinCubicalTopology
-
-/-- The fixed source group has an ordinary three-dimensional classifying CW
-space, with an actually contractible surjective covering. -/
-theorem source_hasClassifyingSpace_three : HasClassifyingSpace.{0} SourceGroup 3 :=
-  ArtinCubicalTopology.height_hasClassifyingSpace seedGraph 3 SeedConnection.clique_card_le_three
-end EilenbergGanea
-
-end
-
-end OAI
+import OAI.AlgebraicTopology.Cubical.Realization
+import OAI.AlgebraicTopology.Cubical.ArtinChains
+import OAI.Topology.EilenbergGanea.Model
+import OAI.Topology.EilenbergGanea.SeedConnection
+import Mathlib.Topology.Covering.Quotient
+import Mathlib.Topology.Covering.Deck
+import Mathlib.Topology.Homotopy.Lifting
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set Topology
+open scoped Quaternion
+
+namespace EilenbergGanea.DescendingCube.Data
+open CubeInterpolation
+variable {G A : Type*} [Group G] {f : A → G} (D : Data f)
+variable [Fintype A]
+
+omit [Fintype A] in
+theorem cellComm (c : D.Cell) : (c.directions : Set A).Pairwise
+    (fun a b => Commute (f a) (f b)) :=
+  fun _ ha _ hb he => D.commute c.top _ _ (c.subset ha) (c.subset hb) he
+
+omit [Fintype A] in
+theorem ones_eq_filter (c : D.Cell) (x : D.Coordinates c) :
+    D.ones c x = c.directions.filter (fun a => D.fullCoord c x a = 1) := by
+  ext a
+  simp only [mem_ones,Finset.mem_filter]
+  constructor
+  · rintro ⟨ha,hx⟩
+    exact ⟨ha,by simpa only [fullCoord,dite_eq_left ha] using hx⟩
+  · rintro ⟨ha,hx⟩
+    exact ⟨ha,by simpa only [fullCoord,dite_eq_left ha] using hx⟩
+
+omit [Fintype A] in
+theorem normalize_fullCoord (c : D.Cell) (x : D.Coordinates c) :
+    CubeInterpolation.normalize (D.fullCoord c x) = D.normalizedCoord c x := by
+  funext a
+  by_cases ha : a ∈ c.directions
+  · simp only [CubeInterpolation.normalize,fullCoord,normalizedCoord,dite_eq_left ha]
+  · simp [CubeInterpolation.normalize,fullCoord,normalizedCoord,ha]
+
+def interpolation (p : D.Point) (φ : G → ℝ) : ℝ :=
+  operator f (D.active p) (D.cellComm (D.pointCell p)) p.coord φ p.top
+
+theorem active_characteristic_subset (c : D.Cell) (x : D.Coordinates c) :
+    D.active (D.characteristic c x) ⊆ c.directions := by
+  intro a ha
+  by_contra h
+  apply ((D.mem_active _ _).mp ha)
+  change D.normalizedCoord c x a = 0
+  simp [normalizedCoord,h]
+
+theorem interpolation_characteristic (c : D.Cell) (x : D.Coordinates c) (φ : G → ℝ) :
+    D.interpolation (D.characteristic c x) φ =
+      operator f c.directions (D.cellComm c) (D.fullCoord c x) φ c.top := by
+  have hsub := D.active_characteristic_subset c x
+  have he := operator_subset f (D.active (D.characteristic c x)) c.directions
+    (D.cellComm c) hsub (D.normalizedCoord c x) (by
+      intro a _ hn
+      exact not_not.mp (by simpa only [mem_active,characteristic] using hn))
+  have hn := operator_normalize f c.directions (D.cellComm c) (D.fullCoord c x)
+  simp only [normalize_fullCoord,← ones_eq_filter] at hn
+  have hh := congrArg (fun T : Module.End ℝ (G → ℝ) => T φ c.top) hn
+  rw [Module.End.mul_apply,translation_apply,he] at hh
+  exact hh.symm
+
+theorem interpolation_continuous (φ : G → ℝ) :
+    Continuous (fun p : D.Point => D.interpolation p φ) := by
+  apply D.evaluation_quotient.continuous_iff.mpr
+  apply continuous_sigma
+  intro c
+  change Continuous (fun x : D.Coordinates c => D.interpolation (D.characteristic c x) φ)
+  simp only [interpolation_characteristic]
+  exact operator_continuous f c.directions _ (D.fullCoord c)
+    (fun a _ => D.fullCoord_continuous c a) φ c.top
+
+theorem interpolation_vertex_pos (p : D.Point) :
+    0 < D.interpolation p (fun a => if a = p.top then 1 else 0) := by
+  apply operator_pos
+  · intro a _; exact ⟨p.nonneg a,p.lt_one a⟩
+  · intro a; split_ifs <;> norm_num
+  · simp
+
+omit [Fintype A] in
+theorem descending_operator_delta_zero (s : Finset A) (g : G) (hs : s ⊆ D.desc g)
+    (x : A → ℝ) (b : G) (hb : D.length g ≤ D.length b) (hne : b ≠ g) :
+    operator f s (fun a ha d hd hne => D.commute g a d (hs ha) (hs hd) hne)
+      x (fun a => if a = b then 1 else 0) g = 0 := by
+  induction s using Finset.induction_on generalizing g with
+  | empty => simp [operator_empty,hne.symm]
+  | @insert a s ha ih =>
+    have hga := hs (Finset.mem_insert_self a s)
+    have hs0 : s ⊆ D.desc g := fun d hd => hs (Finset.mem_insert_of_mem hd)
+    have hs1 : s ⊆ D.desc (g * f a) := by
+      intro d hd
+      exact D.preserve g a d hga (hs0 hd) (by intro he; exact ha (he ▸ hd))
+    have hd := D.decrease g a hga
+    have hb1 : D.length (g*f a) ≤ D.length b := by omega
+    have hne1 : b ≠ g*f a := by intro he; rw [he] at hb; omega
+    have hc : (↑(insert a s) : Set A).Pairwise (fun a d => Commute (f a) (f d)) :=
+      fun _ ha _ hd hne => D.commute g _ _ (hs ha) (hs hd) hne
+    change operator f (insert a s) hc x (fun a => if a = b then 1 else 0) g = 0
+    rw [operator_insert f s a ha,Module.End.mul_apply,blend_apply]
+    rw [ih g hs0 hb hne,ih (g*f a) hs1 hb1 hne1]
+    ring
+
+theorem interpolation_vertex_zero (p : D.Point) (b : G)
+    (hb : D.length p.top ≤ D.length b) (hne : b ≠ p.top) :
+    D.interpolation p (fun a => if a = b then 1 else 0) = 0 :=
+  D.descending_operator_delta_zero (D.active p) p.top (D.pointCell p).subset p.coord b hb hne
+
+theorem interpolation_top_eq (p q : D.Point)
+    (he : ∀ φ, D.interpolation p φ = D.interpolation q φ) : p.top = q.top := by
+  have length_le (p q : D.Point) (he : ∀ φ, D.interpolation p φ = D.interpolation q φ) :
+      D.length q.top ≤ D.length p.top := by
+    by_contra hn
+    have hlt : D.length p.top < D.length q.top := lt_of_not_ge hn
+    have hne : q.top ≠ p.top := by intro eq; rw [eq] at hlt; exact lt_irrefl _ hlt
+    have hz := D.interpolation_vertex_zero p q.top hlt.le hne
+    have hp := D.interpolation_vertex_pos q
+    rw [← he,hz] at hp
+    exact lt_irrefl _ hp
+  have h₁ := length_le p q he
+  by_contra hne
+  have hz := D.interpolation_vertex_zero q p.top h₁ hne
+  have hp := D.interpolation_vertex_pos p
+  rw [he,hz] at hp
+  exact lt_irrefl _ hp
+
+variable (exponent : A → G →* Multiplicative ℝ)
+variable (hexponent : ∀ g a, a ∈ D.desc g → ∀ b, b ∈ D.desc g →
+  (exponent a (f b)).toAdd = if a = b then 1 else 0)
+
+include hexponent in
+theorem interpolation_exponent (p : D.Point) (a : A) (ha : a ∈ D.desc p.top) :
+    D.interpolation p (fun g => (exponent a g).toAdd) =
+      (exponent a p.top).toAdd + p.coord a := by
+  refine (operator_linear f (exponent a) (D.active p)
+    (D.cellComm (D.pointCell p)) p.coord p.top).trans ?_
+  congr 1
+  have he : ∀ b ∈ D.active p, (exponent a (f b)).toAdd = if a = b then 1 else 0 :=
+    fun b hb => hexponent p.top a ha b ((D.pointCell p).subset hb)
+  simp only [Finset.sum_congr rfl (fun b hb => congrArg (fun t : ℝ => p.coord b * t) (he b hb)),
+    mul_ite,mul_one,mul_zero,Finset.sum_ite_eq]
+  by_cases h : p.coord a = 0
+  · simp [h]
+  · simp [mem_active,h]
+
+include exponent hexponent in
+theorem interpolation_injective : Function.Injective
+    (fun p : D.Point => fun φ : G → ℝ => D.interpolation p φ) := by
+  intro p q he
+  have he' : ∀ φ, D.interpolation p φ = D.interpolation q φ := fun φ => congrFun he φ
+  have hb := D.interpolation_top_eq p q he'
+  apply Point.ext hb
+  funext a
+  by_cases ha : a ∈ D.desc p.top
+  · have hq : a ∈ D.desc q.top := by simpa only [hb] using ha
+    have hh := he' (fun g => (exponent a g).toAdd)
+    rw [D.interpolation_exponent exponent hexponent p a ha,
+      D.interpolation_exponent exponent hexponent q a hq,hb] at hh
+    exact add_left_cancel hh
+  · have hp : p.coord a = 0 := by_contra (fun hn => ha (p.support a hn))
+    have hq : q.coord a = 0 := by
+      by_contra hn
+      exact ha (by simpa only [hb] using q.support a hn)
+    rw [hp,hq]
+
+include exponent hexponent in
+theorem point_t2 : T2Space D.Point :=
+  T2Space.of_injective_continuous
+    (D.interpolation_injective exponent hexponent)
+    (continuous_pi (fun φ => D.interpolation_continuous φ))
+
+end EilenbergGanea.DescendingCube.Data
+
+open Classical Topology
+namespace WeakRealization
+
+/-- A homeomorphism is induced by point-separating coordinates and actual
+continuous lifts on each characteristic disk, in both directions. -/
+def homeomorph {ι κ X Y Z : Type*} {E : ι → Type*} {F : κ → Type*}
+    [TopologicalSpace X] [TopologicalSpace Y]
+    [∀ i, TopologicalSpace (E i)] [∀ j, TopologicalSpace (F j)]
+    (e : ((i : ι) × E i) → X) (f : ((j : κ) × F j) → Y)
+    (he : IsQuotientMap e) (hf : IsQuotientMap f)
+    (J : X → Z) (K : Y → Z) (hJ : Function.Injective J) (hK : Function.Injective K)
+    (hxy : ∀ i, ∃ φ : C(E i,Y), ∀ t, K (φ t) = J (e ⟨i,t⟩))
+    (hyx : ∀ j, ∃ φ : C(F j,X), ∀ t, J (φ t) = K (f ⟨j,t⟩)) : X ≃ₜ Y := by
+  have hx : ∀ x, ∃ y, K y = J x := by
+    intro x
+    obtain ⟨⟨i,t⟩,rfl⟩ := he.surjective x
+    obtain ⟨φ,hφ⟩ := hxy i
+    exact ⟨φ t,hφ t⟩
+  have hy : ∀ y, ∃ x, J x = K y := by
+    intro y
+    obtain ⟨⟨j,t⟩,rfl⟩ := hf.surjective y
+    obtain ⟨φ,hφ⟩ := hyx j
+    exact ⟨φ t,hφ t⟩
+  let A : X → Y := fun x => (hx x).choose
+  let B : Y → X := fun y => (hy y).choose
+  have hA : ∀ x, K (A x) = J x := fun x => (hx x).choose_spec
+  have hB : ∀ y, J (B y) = K y := fun y => (hy y).choose_spec
+  refine {
+    toEquiv := {
+      toFun := A
+      invFun := B
+      left_inv := fun x => hJ ((hB (A x)).trans (hA x))
+      right_inv := fun y => hK ((hA (B y)).trans (hB y)) }
+    continuous_toFun := ?_
+    continuous_invFun := ?_ }
+  · apply he.continuous_iff.mpr
+    apply continuous_sigma
+    intro i
+    obtain ⟨φ,hφ⟩ := hxy i
+    have hh : (fun t => A (e ⟨i,t⟩)) = φ := by
+      funext t
+      exact hK ((hA _).trans (hφ t).symm)
+    change Continuous (fun t => A (e ⟨i,t⟩))
+    rw [hh]
+    exact φ.continuous
+  · apply hf.continuous_iff.mpr
+    apply continuous_sigma
+    intro j
+    obtain ⟨φ,hφ⟩ := hyx j
+    have hh : (fun t => B (f ⟨j,t⟩)) = φ := by
+      funext t
+      exact hJ ((hB _).trans (hφ t).symm)
+    change Continuous (fun t => B (f ⟨j,t⟩))
+    rw [hh]
+    exact φ.continuous
+
+end WeakRealization
+
+namespace EilenbergGanea.CubeInterpolation
+open CubicalChains.Reorientation
+variable {G V : Type*} [Group G]
+
+/-- Interpolation in a listed cube, independent of any topological model. -/
+def listed (f : V → G) (w : List V) (x : V → ℝ) : Module.End ℝ (G → ℝ) :=
+  (w.map (fun v => blend (f v) (x v))).prod
+
+@[simp] theorem listed_nil (f : V → G) (x : V → ℝ) : listed f [] x = 1 := rfl
+@[simp] theorem listed_cons (f : V → G) (v : V) (w : List V) (x : V → ℝ) :
+    listed f (v::w) x = blend (f v) (x v) * listed f w x := rfl
+
+theorem operator_listed [DecidableEq V] (f : V → G) (w : List V) (hw : w.Nodup)
+    (hc : (w.toFinset : Set V).Pairwise (fun v u => Commute (f v) (f u))) (x : V → ℝ) :
+    operator f w.toFinset hc x = listed f w x :=
+  Finset.noncommProd_toFinset w _ _ hw
+
+/-- Literal multilinear invariance under reversing selected cube directions. -/
+theorem listed_reorientation (f : V → G) (w : List (Signed V))
+    (hw : w.Pairwise (fun a b => Commute (f a.1) (f b.1)))
+    (x : Signed V → ℝ) (z : V → ℝ)
+    (hx : ∀ a ∈ w, x a = if a.2 then 1-z a.1 else z a.1) (φ : G → ℝ) (g : G) :
+    listed (step f) w x φ g = listed f (w.map Prod.fst) z φ (origin f g w) := by
+  induction w generalizing g with
+  | nil => rfl
+  | cons a w ih =>
+    obtain ⟨ha,hw⟩ := List.pairwise_cons.mp hw
+    have hxw : ∀ b ∈ w, x b = if b.2 then 1-z b.1 else z b.1 :=
+      fun b hb => hx b (by simp [hb])
+    simp only [listed_cons,List.map_cons,Module.End.mul_apply,blend_apply,
+      ih hw hxw,hx a (by simp),step,origin]
+    cases hb : a.2
+    · simp only [Bool.false_eq_true,↓reduceIte]
+      rw [origin_shift f g (f a.1) w ha]
+    · simp only [↓reduceIte]
+      rw [origin_shift f g (f a.1)⁻¹ w (fun b hb => (ha b hb).inv_left),
+        inv_mul_cancel_right]
+      ring
+
+end EilenbergGanea.CubeInterpolation
+
+namespace EilenbergGanea.ArtinCubicalTopology
+open TraceWords CubeInterpolation CubicalChains.Reorientation
+variable {V : Type*} [Fintype V] (L : SimpleGraph V)
+
+local instance : Std.Symm L.Adj := ⟨fun _ _ h => h.symm⟩
+local instance : Std.Irrefl L.Adj := ⟨fun _ => L.irrefl⟩
+
+abbrev f := artinGenerator L
+abbrev comm := fun v w h => adjacent_generators_commute L (v := v) (w := w) h
+abbrev D := artinDescendingData L
+abbrev Positive := CubeRealization.Point L (f L) (comm L)
+abbrev Descending := (D L).Point
+
+omit [Fintype V] in
+@[simp] theorem step_eq (a : Letter V) : step (f L) a = letterValue (f L) (invLetter a) := by
+  cases a with | mk v b => cases b <;> rfl
+
+def exponent (v : V) : ArtinGroup L →* Multiplicative ℝ :=
+  artinLift L (fun w => Multiplicative.ofAdd (if v = w then (1 : ℝ) else 0))
+    (fun _ _ _ => Commute.all _ _)
+
+omit [Fintype V] in
+@[simp] theorem exponent_generator (v w : V) :
+    (exponent L v (f L w)).toAdd = if v = w then 1 else 0 := by
+  simp [exponent,f]
+
+omit [Fintype V] in
+@[simp] theorem height_generator (v : V) : (height L (f L v)).toAdd = 1 := by
+  exact congrArg Multiplicative.toAdd (PresentedGroup.toGroup.of _)
+
+def signedExponent (a : Letter V) : ArtinGroup L →* Multiplicative ℝ :=
+  artinLift L (fun v => Multiplicative.ofAdd
+    (if a.1 = v then (if a.2 then (-1 : ℝ) else 1) else 0))
+    (fun _ _ _ => Commute.all _ _)
+
+theorem signedExponent_step (g : ArtinGroup L) (a : Letter V) (ha : a ∈ (D L).desc g)
+    (b : Letter V) (hb : b ∈ (D L).desc g) :
+    (signedExponent L a (letterValue (f L) (invLetter b))).toAdd = if a = b then 1 else 0 := by
+  by_cases he : a = b
+  · subst b
+    rcases a with ⟨v,b⟩
+    cases b <;> simp [signedExponent,letterValue,invLetter,f]
+  · have hi : Independent L a b := rightDescents_pairwise L g
+      (by simpa [D,artinDescendingData] using ha) (by simpa [D,artinDescendingData] using hb) he
+    have hv : a.1 ≠ b.1 := hi.ne
+    rcases b with ⟨v,b⟩
+    cases b <;> simp [signedExponent,letterValue,invLetter,f,hv,he]
+
+theorem positive_interp_inj : Function.Injective
+    (fun p : Positive L => fun φ => CubeRealization.interpolation L (f L) (comm L) p φ) :=
+  CubeRealization.interpolation_injective L (f L) (comm L) (height L) (height_generator L)
+    (exponent L) (exponent_generator L)
+
+theorem descending_interp_inj : Function.Injective
+    (fun p : Descending L => fun φ => (D L).interpolation p φ) :=
+  (D L).interpolation_injective (signedExponent L) (by
+    intro g a ha b hb
+    by_cases he : a = b <;> simpa [he] using signedExponent_step L g a ha b hb)
+
+def flip (b : Bool) (t : unitInterval) : unitInterval :=
+  if b then unitInterval.symm t else t
+
+@[simp] theorem flip_flip (b : Bool) (t : unitInterval) : flip b (flip b t) = t := by
+  cases b <;> simp [flip]
+
+theorem flip_continuous (b : Bool) : Continuous (flip b) := by
+  cases b
+  · exact continuous_id
+  · exact unitInterval.continuous_symm
+
+def vertexEquiv (w : Word V) (hw : (w.map Prod.fst).Nodup) :
+    w.toFinset ≃ (w.map Prod.fst).toFinset :=
+  Equiv.ofBijective (fun a => ⟨a.val.1,by
+    exact List.mem_toFinset.mpr (List.mem_map.mpr ⟨a.val,List.mem_toFinset.mp a.property,rfl⟩)⟩) (by
+    constructor
+    · intro a b he
+      apply Subtype.ext
+      exact (List.nodup_map_iff_inj_on (hw.of_map Prod.fst)).mp hw
+        a.val (List.mem_toFinset.mp a.property) b.val (List.mem_toFinset.mp b.property) (congrArg Subtype.val he)
+    · intro v
+      obtain ⟨a,ha,he⟩ := List.mem_map.mp (List.mem_toFinset.mp v.property)
+      exact ⟨⟨a,by simpa using ha⟩,Subtype.ext he⟩)
+
+omit [Fintype V] in
+@[simp] theorem vertexEquiv_val (w : Word V) (hw : (w.map Prod.fst).Nodup)
+    (a : w.toFinset) : (vertexEquiv w hw a).val = a.val.1 := rfl
+
+def toPositiveCoords (w : Word V) (hw : (w.map Prod.fst).Nodup)
+    (x : w.toFinset → unitInterval) : (w.map Prod.fst).toFinset → unitInterval :=
+  fun v => flip ((vertexEquiv w hw).symm v).val.2 (x ((vertexEquiv w hw).symm v))
+
+omit [Fintype V] in
+@[simp] theorem toPositiveCoords_apply (w : Word V) (hw : (w.map Prod.fst).Nodup)
+    (x : w.toFinset → unitInterval) (a : w.toFinset) :
+    toPositiveCoords w hw x (vertexEquiv w hw a) = flip a.val.2 (x a) := by
+  simp [toPositiveCoords]
+
+omit [Fintype V] in
+theorem toPositiveCoords_continuous (w : Word V) (hw : (w.map Prod.fst).Nodup) :
+    Continuous (toPositiveCoords w hw) :=
+  continuous_pi (fun _ => (flip_continuous _).comp (continuous_apply _))
+
+def toDescendingCoords (w : Word V) (hw : (w.map Prod.fst).Nodup)
+    (z : (w.map Prod.fst).toFinset → unitInterval) : w.toFinset → unitInterval :=
+  fun a => flip a.val.2 (z (vertexEquiv w hw a))
+
+omit [Fintype V] in
+theorem toDescendingCoords_continuous (w : Word V) (hw : (w.map Prod.fst).Nodup) :
+    Continuous (toDescendingCoords w hw) :=
+  continuous_pi (fun _ => (flip_continuous _).comp (continuous_apply _))
+
+omit [Fintype V] in
+@[simp] theorem toPositive_toDescending (w : Word V) (hw : (w.map Prod.fst).Nodup)
+    (z : (w.map Prod.fst).toFinset → unitInterval) :
+    toPositiveCoords w hw (toDescendingCoords w hw z) = z := by
+  funext v
+  obtain ⟨a,rfl⟩ := (vertexEquiv w hw).surjective v
+  simp [toDescendingCoords]
+
+def positiveCell (g : ArtinGroup L) (w : Word V) (hw : (w.map Prod.fst).Pairwise L.Adj) :
+    CubeRealization.Cell (G := ArtinGroup L) L where
+  base := origin (f L) g w
+  directions := (w.map Prod.fst).toFinset
+  clique := fun _ ha _ hb he => hw.forall (List.mem_toFinset.mp ha) (List.mem_toFinset.mp hb) he
+
+def descendingCell (g : ArtinGroup L) (w : Word V)
+    (hd : ∀ a ∈ w, a ∈ rightDescents L g) : (D L).Cell where
+  top := g
+  directions := w.toFinset
+  subset a ha := by simpa [D,artinDescendingData] using hd a (List.mem_toFinset.mp ha)
+
+theorem interpolation_reorientation (g : ArtinGroup L) (w : Word V)
+    (hw : (w.map Prod.fst).Pairwise L.Adj) (hd : ∀ a ∈ w, a ∈ rightDescents L g)
+    (x : (D L).Coordinates (descendingCell L g w hd)) (φ : ArtinGroup L → ℝ) :
+    (D L).interpolation ((D L).characteristic (descendingCell L g w hd) x) φ =
+      CubeRealization.interpolation L (f L) (comm L)
+        (CubeRealization.characteristic (f L) (comm L) (positiveCell L g w hw)
+          (toPositiveCoords w hw.nodup x)) φ := by
+  refine ((D L).interpolation_characteristic (descendingCell L g w hd) x φ).trans ?_
+  refine Eq.trans ?_ (CubeRealization.interpolation_characteristic L (f L) (comm L)
+    (positiveCell L g w hw) (toPositiveCoords w hw.nodup x) φ).symm
+  have hf : (fun a : Letter V => letterValue (artinGenerator L) (invLetter a)) = step (f L) :=
+    funext (fun a => (step_eq L a).symm)
+  erw [operator_listed _ w (hw.nodup.of_map Prod.fst),operator_listed _ _ hw.nodup]
+  have hop := congrArg (fun k : Letter V → ArtinGroup L => listed k w) hf
+  rw [hop]
+  apply listed_reorientation
+  · exact (List.pairwise_map.mp hw).imp (fun hh => adjacent_generators_commute L hh)
+  · intro a ha
+    let aa : w.toFinset := ⟨a,by simpa using ha⟩
+    have h₁ : (D L).fullCoord (descendingCell L g w hd) x a = (x aa : ℝ) := by
+      exact (D L).fullCoord_mem (descendingCell L g w hd) x aa
+    have h₂ : CubeRealization.fullCoord (positiveCell L g w hw)
+        (toPositiveCoords w hw.nodup x) a.1 = (flip a.2 (x aa) : ℝ) := by
+      have hh := CubeRealization.fullCoord_mem (positiveCell L g w hw)
+        (toPositiveCoords w hw.nodup x) (vertexEquiv w hw.nodup aa)
+      exact hh.trans (congrArg (fun z : unitInterval => (z : ℝ))
+        (toPositiveCoords_apply w hw.nodup x aa))
+    rw [h₁,h₂]
+    cases h : a.2 <;> simp [flip]
+
+theorem descendingCell_rep (c : (D L).Cell) :
+    ∃ (g : ArtinGroup L) (w : Word V) (_hw : (w.map Prod.fst).Pairwise L.Adj)
+      (hd : ∀ a ∈ w, a ∈ rightDescents L g), descendingCell L g w hd = c := by
+  let w := c.directions.toList
+  have hd : ∀ a ∈ w, a ∈ rightDescents L c.top := by
+    intro a ha
+    have hh := c.subset (Finset.mem_toList.mp ha)
+    simpa [D,artinDescendingData] using hh
+  have hw : (w.map Prod.fst).Pairwise L.Adj := by
+    apply List.pairwise_map.mpr
+    apply c.directions.nodup_toList.pairwise_of_forall_ne
+    intro a ha b hb he
+    exact rightDescents_pairwise L c.top (hd a ha) (hd b hb) he
+  refine ⟨c.top,w,hw,hd,?_⟩
+  cases c with
+  | mk g s hs =>
+    dsimp only [descendingCell,w]
+    congr
+    exact s.toList_toFinset
+
+omit [Fintype V] in
+theorem positiveCell_rep (c : CubeRealization.Cell (G := ArtinGroup L) L) :
+    ∃ (g : ArtinGroup L) (w : Word V) (hw : (w.map Prod.fst).Pairwise L.Adj)
+      (_hd : ∀ a ∈ w, a ∈ rightDescents L g), positiveCell L g w hw = c := by
+  let v := c.directions.toList
+  have hv : v.Pairwise L.Adj := c.directions.nodup_toList.pairwise_of_forall_ne
+    (fun a ha b hb he => c.clique (Finset.mem_toList.mp ha) (Finset.mem_toList.mp hb) he)
+  let o := CubeDescents.orient L c.base v
+  have hw : (o.2.map Prod.fst).Pairwise L.Adj := by
+    simpa only [o,CubeDescents.orient_map] using hv
+  have hd : ∀ a ∈ o.2, a ∈ rightDescents L o.1 := CubeDescents.orient_descents L c.base v hv
+  refine ⟨o.1,o.2,hw,hd,?_⟩
+  have hb : origin (f L) o.1 o.2 = c.base := CubeDescents.origin_orient L c.base v hv
+  have hs : (o.2.map Prod.fst).toFinset = c.directions := by
+    simp only [o,CubeDescents.orient_map,v,Finset.toList_toFinset]
+  cases c with
+  | mk g s hc =>
+    change CubeRealization.Cell.mk (origin (f L) o.1 o.2) (o.2.map Prod.fst).toFinset _ = _
+    congr
+
+theorem descending_local_lift (c : (D L).Cell) :
+    ∃ φ : C((D L).Coordinates c,Positive L), ∀ x,
+      (fun ψ => CubeRealization.interpolation L (f L) (comm L) (φ x) ψ) =
+        (fun ψ => (D L).interpolation ((D L).characteristic c x) ψ) := by
+  obtain ⟨g,w,hw,hd,rfl⟩ := descendingCell_rep L c
+  refine ⟨⟨fun x => CubeRealization.characteristic (f L) (comm L) (positiveCell L g w hw)
+    (toPositiveCoords w hw.nodup x),
+    (CubeRealization.characteristic_continuous (f L) (comm L) (positiveCell L g w hw)).comp
+      (toPositiveCoords_continuous w hw.nodup)⟩,?_⟩
+  intro x
+  funext ψ
+  exact (interpolation_reorientation L g w hw hd x ψ).symm
+
+theorem positive_local_lift (c : CubeRealization.Cell (G := ArtinGroup L) L) :
+    ∃ φ : C(CubeRealization.Coordinates c,Descending L), ∀ x,
+      (fun ψ => (D L).interpolation (φ x) ψ) =
+        (fun ψ => CubeRealization.interpolation L (f L) (comm L)
+          (CubeRealization.characteristic (f L) (comm L) c x) ψ) := by
+  obtain ⟨g,w,hw,hd,rfl⟩ := positiveCell_rep L c
+  refine ⟨⟨fun x => (D L).characteristic (descendingCell L g w hd) (toDescendingCoords w hw.nodup x),
+    ((D L).characteristic_continuous (descendingCell L g w hd)).comp
+      (toDescendingCoords_continuous w hw.nodup)⟩,?_⟩
+  intro x
+  funext ψ
+  change (D L).interpolation ((D L).characteristic (descendingCell L g w hd)
+    (toDescendingCoords w hw.nodup x)) ψ = _
+  change (w.map Prod.fst).toFinset → unitInterval at x
+  simpa only [toPositive_toDescending] using interpolation_reorientation L g w hw hd
+    (toDescendingCoords w hw.nodup x) ψ
+
+/-- Reorientation preserves the actual weak topologies, not only cellular chains. -/
+def reorient : Descending L ≃ₜ Positive L :=
+  WeakRealization.homeomorph (D L).evaluation (CubeRealization.evaluation (f L) (comm L))
+    (D L).evaluation_quotient (CubeRealization.evaluation_quotient (f L) (comm L))
+    (fun p ψ => (D L).interpolation p ψ)
+    (fun p ψ => CubeRealization.interpolation L (f L) (comm L) p ψ)
+    (descending_interp_inj L) (positive_interp_inj L)
+    (descending_local_lift L) (positive_local_lift L)
+
+variable [Nonempty V]
+
+/-- The ordinary positive-cube CW realization is genuinely contractible. -/
+theorem positive_contractible : ContractibleSpace (Positive L) := by
+  let : ContractibleSpace (Descending L) := artinDescending_contractible L
+  exact (reorient L).symm.contractibleSpace
+
+end EilenbergGanea.ArtinCubicalTopology
+
+open Classical Set Topology
+namespace CompactChart
+
+/-- A characteristic map which has singleton fibers over its open cell has a
+continuous inverse there. Compactness is used only on the closed disk; no
+regularity of the attaching sphere is required. -/
+theorem continuousOn_symm {A X : Type*} [TopologicalSpace A] [TopologicalSpace X]
+    [T2Space X] (e : PartialEquiv A X) (K : Set A) (hK : IsCompact K)
+    (hsub : e.source ⊆ K) (hc : ContinuousOn e K)
+    (hboundary : ∀ a ∈ K, e a ∈ e.target → a ∈ e.source) :
+    ContinuousOn e.symm e.target := by
+  let : CompactSpace K := isCompact_iff_compactSpace.mp hK
+  let f : K → X := fun a => e a
+  have hf : Continuous f := continuousOn_iff_continuous_domRestrict.mp hc
+  let r := e.target.restrictPreimage f
+  have hr : Function.Surjective r := by
+    intro x
+    refine ⟨⟨⟨e.symm x,hsub (e.map_target x.property)⟩,?_⟩,?_⟩
+    · change e (e.symm x) ∈ e.target
+      rw [e.right_inv x.property]
+      exact x.property
+    · exact Subtype.ext (e.right_inv x.property)
+  have hq : IsQuotientMap r :=
+    (hf.isClosedMap.restrictPreimage e.target).isQuotientMap hf.restrictPreimage hr
+  apply continuousOn_iff_continuous_domRestrict.mpr
+  apply hq.continuous_iff.mpr
+  have he : (fun x : e.target => e.symm x) ∘ r =
+      (fun a : f ⁻¹' e.target => (a.val.val : A)) := by
+    funext a
+    exact e.left_inv (hboundary a.val a.val.property a.property)
+  change Continuous ((fun x : e.target => e.symm x) ∘ r)
+  rw [he]
+  exact continuous_subtype_val.comp continuous_subtype_val
+
+end CompactChart
+
+namespace EilenbergGanea.CubeRealization
+
+section
+open Metric
+variable {G V : Type*} [Group G] [Fintype V]
+variable {L : SimpleGraph V} (f : V → G)
+variable (comm : ∀ v w, L.Adj v w → Commute (f v) (f w))
+
+theorem chartInverse_continuousOn_of_t2 [T2Space (Point (G := G) L f comm)]
+    {n : ℕ} (c : DimCell (G := G) (L := L) n) :
+    ContinuousOn (chart f comm c).symm (chart f comm c).target := by
+  apply CompactChart.continuousOn_symm (chart f comm c) (closedBall 0 1)
+    (isCompact_closedBall _ _) ball_subset_closedBall (chart_continuous f comm c).continuousOn
+  intro t ht he
+  change pointCell (chart f comm c t) = c.val at he
+  change t ∈ ball 0 1
+  by_contra hn
+  have hs : t ∈ sphere 0 1 := mem_sphere.mpr
+    (le_antisymm (mem_closedBall.mp ht) (le_of_not_gt hn))
+  have hd := boundary_dimension_lt f comm c hs
+  rw [he,c.property] at hd
+  exact (lt_irrefl _) hd
+
+/-- The ordinary CW structure follows from the genuine normalized-cube gluing;
+only Hausdorffness and compact characteristic disks are used. -/
+@[instance_reducible] def cwComplexOfT2 [T2Space (Point (G := G) L f comm)] : CWComplex (univ : Set (Point (G := G) L f comm)) where
+  cell := DimCell (G := G) (L := L)
+  map := fun _ c => chart f comm c
+  source_eq := fun _ _ => rfl
+  continuousOn := fun _ c => (chart_continuous f comm c).continuousOn
+  continuousOn_symm := fun _ c => chartInverse_continuousOn_of_t2 f comm c
+  pairwiseDisjoint' := by
+    intro a _ b _ hab
+    change Disjoint (chart f comm a.2 '' ball 0 1) (chart f comm b.2 '' ball 0 1)
+    rw [chart_image_ball,chart_image_ball]
+    apply Set.disjoint_left.mpr
+    intro p ha hb
+    apply hab
+    have he : a.2.val = b.2.val := ha.symm.trans hb
+    have hn : a.1 = b.1 := a.2.property.symm.trans ((congrArg (fun c => c.directions.card) he).trans b.2.property)
+    cases a with | mk n a =>
+      cases b with | mk m b =>
+        dsimp at hn
+        subst m
+        exact congrArg (Sigma.mk n) (Subtype.ext he)
+  mapsTo' := chart_mapsTo f comm
+  closed' := by
+    intro A _ hA
+    apply (evaluation_quotient f comm).isClosed_preimage.mp
+    rw [isClosed_sigma_iff]
+    intro c
+    let d : DimCell (G := G) (L := L) c.directions.card := ⟨c,rfl⟩
+    have hh := (hA _ d).preimage (characteristic_continuous f comm c)
+    rw [chart_image_disk] at hh
+    change IsClosed (characteristic f comm c ⁻¹' (A ∩ Set.range (characteristic f comm c))) at hh
+    change IsClosed (characteristic f comm c ⁻¹' A)
+    simpa only [Set.preimage_inter,Set.preimage_range,Set.inter_univ] using hh
+  union' := by
+    apply Set.eq_univ_of_forall
+    intro p
+    let d : DimCell (G := G) (L := L) (pointCell p).directions.card := ⟨pointCell p,rfl⟩
+    apply Set.mem_iUnion.mpr
+    refine ⟨_,Set.mem_iUnion.mpr ⟨d,?_⟩⟩
+    rw [chart_image_disk]
+    exact ⟨pointCoordinates p,characteristic_point f comm p⟩
+
+end
+
+variable {G Q V : Type*} [Group G] [Group Q] [Fintype V]
+variable {L : SimpleGraph V} (f : V → G)
+variable (comm : ∀ v w, L.Adj v w → Commute (f v) (f w))
+variable (q : G →* Q)
+
+include comm in
+omit [Fintype V] in
+theorem mappedComm : ∀ v w, L.Adj v w → Commute ((q ∘ f) v) ((q ∘ f) w) :=
+  fun v w hvw => (comm v w hvw).map q
+
+/-- Applying a group homomorphism only to the initial vertices of cubes. -/
+def mapPoint (p : Point (G := G) L f comm) : Point (G := Q) L (q ∘ f) (mappedComm f comm q) where
+  base := q p.base
+  coord := p.coord
+  nonneg := p.nonneg
+  lt_one := p.lt_one
+  clique := p.clique
+
+def mapCell (c : Cell (G := G) L) : Cell (G := Q) L where
+  base := q c.base
+  directions := c.directions
+  clique := c.clique
+
+omit [Fintype V] in
+@[simp] theorem mapPoint_base (p : Point (G := G) L f comm) :
+    (mapPoint f comm q p).base = q p.base := rfl
+
+omit [Fintype V] in
+@[simp] theorem mapPoint_coord (p : Point (G := G) L f comm) :
+    (mapPoint f comm q p).coord = p.coord := rfl
+
+omit [Fintype V] in
+@[simp] theorem mapPoint_characteristic (c : Cell (G := G) L) (x : Coordinates c) :
+    mapPoint f comm q (characteristic f comm c x) =
+      characteristic (q ∘ f) (mappedComm f comm q) (mapCell q c) x := by
+  apply Point.ext
+  · change q (c.base * product f comm (ones c x) _) =
+      q c.base * product (q ∘ f) (mappedComm f comm q) (ones c x) _
+    rw [map_mul]
+    congr 1
+    exact Finset.map_noncommProd _ _ _ q
+  · rfl
+
+theorem mapPoint_continuous : Continuous (mapPoint f comm q) := by
+  apply (evaluation_quotient f comm).continuous_iff.mpr
+  apply continuous_sigma
+  intro c
+  change Continuous (fun x => mapPoint f comm q (characteristic f comm c x))
+  simp only [mapPoint_characteristic]
+  exact characteristic_continuous (q ∘ f) (mappedComm f comm q) (mapCell q c)
+
+omit [Fintype V] in
+theorem mapCell_surjective (hq : Function.Surjective q) : Function.Surjective (mapCell (L := L) q) := by
+  rintro ⟨a,s,hs⟩
+  obtain ⟨g,rfl⟩ := hq a
+  exact ⟨⟨g,s,hs⟩,rfl⟩
+
+omit [Fintype V] in
+theorem mapPoint_surjective (hq : Function.Surjective q) : Function.Surjective (mapPoint f comm q) := by
+  rintro ⟨a,x,hx,ht,hc⟩
+  obtain ⟨g,rfl⟩ := hq a
+  exact ⟨⟨g,x,hx,ht,hc⟩,rfl⟩
+
+/-- The base-label quotient is a genuine topological quotient for the weak
+cubical topologies, proved by lifting each actual characteristic disk. -/
+theorem mapPoint_quotient (hq : Function.Surjective q) : IsQuotientMap (mapPoint f comm q) := by
+  apply isQuotientMap_iff_isClosed.mpr
+  refine ⟨mapPoint_surjective f comm q hq,?_⟩
+  intro A
+  constructor
+  · exact fun hA => hA.preimage (mapPoint_continuous f comm q)
+  · intro hA
+    apply (evaluation_quotient (q ∘ f) (mappedComm f comm q)).isClosed_preimage.mp
+    rw [isClosed_sigma_iff]
+    intro c
+    obtain ⟨d,rfl⟩ := mapCell_surjective q hq c
+    have hh := hA.preimage (characteristic_continuous f comm d)
+    change IsClosed ((fun x => mapPoint f comm q (characteristic f comm d x)) ⁻¹' A) at hh
+    change IsClosed ((fun x : Coordinates d =>
+      characteristic (q ∘ f) (mappedComm f comm q) (mapCell q d) x) ⁻¹' A)
+    simpa only [mapPoint_characteristic] using hh
+
+omit [Fintype V] in
+/-- The literal group-kernel orbits are exactly the fibers of the cube map. -/
+theorem mapPoint_eq_iff_orbit (p r : Point (G := G) L f comm) :
+    mapPoint f comm q p = mapPoint f comm q r ↔ p ∈ MulAction.orbit q.ker r := by
+  constructor
+  · intro he
+    have hb : q p.base = q r.base := congrArg Point.base he
+    have hx : p.coord = r.coord := congrArg (fun z : Point (G := Q) L (q ∘ f)
+      (mappedComm f comm q) => z.coord) he
+    let g : q.ker := ⟨p.base * r.base⁻¹,by simp only [MonoidHom.mem_ker,map_mul,map_inv,hb,mul_inv_cancel]⟩
+    refine ⟨g,?_⟩
+    apply Point.ext
+    · change (p.base * r.base⁻¹) * r.base = p.base
+      simp [mul_assoc]
+    · exact hx.symm
+  · rintro ⟨g,rfl⟩
+    apply Point.ext
+    · change q ((g : G) * r.base) = q r.base
+      rw [map_mul,g.property,one_mul]
+    · rfl
+
+/-- The quotient homeomorphism respects the actual weak topologies. -/
+def kernelQuotientHomeomorph (hq : Function.Surjective q) :
+    Quotient (MulAction.orbitRel q.ker (Point (G := G) L f comm)) ≃ₜ
+      Point (G := Q) L (q ∘ f) (mappedComm f comm q) :=
+  (Homeomorph.Quotient.congrRight (fun p r => (mapPoint_eq_iff_orbit f comm q p r).symm)).trans
+    (Topology.IsQuotientMap.homeomorph (f := ⟨mapPoint f comm q,mapPoint_continuous f comm q⟩)
+      (mapPoint_quotient f comm q hq))
+
+variable (h : G →* Multiplicative ℤ) (hf : ∀ v, (h (f v)).toAdd = 1)
+variable (exponent : V → G →* Multiplicative ℝ)
+variable (hexponent : ∀ v w, (exponent v (f w)).toAdd = if v = w then 1 else 0)
+
+include h hf exponent hexponent in
+theorem quotientPoint_t2 (hq : Function.Surjective q) :
+    T2Space (Point (G := Q) L (q ∘ f) (mappedComm f comm q)) := by
+  let := point_t2 L f comm h hf exponent hexponent
+  let := point_weaklyLocallyCompact L f comm
+  exact (kernelQuotientHomeomorph f comm q hq).t2Space
+
+include h hf exponent hexponent in
+theorem mapPoint_covering (hq : Function.Surjective q) :
+    IsQuotientCoveringMap (mapPoint f comm q) q.ker where
+  toIsQuotientMap := mapPoint_quotient f comm q hq
+  continuous_const_smul := continuous_const_smul
+  apply_eq_iff_mem_orbit := mapPoint_eq_iff_orbit f comm q _ _
+  disjoint := (subgroup_quotient_covering f comm h hf exponent hexponent q.ker).disjoint
+
+end EilenbergGanea.CubeRealization
+
+namespace EilenbergGanea
+universe u
+namespace ArtinCubicalTopology
+open CubeRealization
+variable {V : Type u} [Fintype V] [Nonempty V] (L : SimpleGraph V)
+
+omit [Fintype V] in
+theorem height_surjective : Function.Surjective (height L) := by
+  intro n
+  let v : V := Classical.choice inferInstance
+  refine ⟨f L v ^ n.toAdd,?_⟩
+  rw [map_zpow]
+  apply Multiplicative.toAdd.injective
+  change n.toAdd • (height L (f L v)).toAdd = n.toAdd
+  simp only [height_generator,zsmul_eq_mul,mul_one,Int.cast_id]
+
+def HeightQuotient := Point L ((height L) ∘ f L) (mappedComm (f L) (comm L) (height L))
+
+instance : TopologicalSpace (HeightQuotient L) := inferInstanceAs (TopologicalSpace
+  (Point L ((height L) ∘ f L) (mappedComm (f L) (comm L) (height L))))
+
+/-- An ordinary cubical classifying space for the height kernel, with one cube
+for every integral level and clique. Contractibility above is the explicit
+word-length contraction, and the fundamental-group identification reuses
+Mathlib's checked covering-monodromy equivalence. -/
+theorem height_hasClassifyingSpace (d : ℕ)
+    (hd : ∀ s : Finset V, (s : Set V).Pairwise L.Adj → s.card ≤ d) :
+    HasClassifyingSpace.{u} (height L).ker d := by
+  let X := Point L ((height L) ∘ f L) (mappedComm (f L) (comm L) (height L))
+  let E := Positive L
+  let : ContractibleSpace E := positive_contractible L
+  let : T2Space X := quotientPoint_t2 (f L) (comm L) (height L)
+    (height L) (height_generator L) (exponent L) (exponent_generator L) (height_surjective L)
+  let : Topology.CWComplex (Set.univ : Set X) :=
+    cwComplexOfT2 ((height L) ∘ f L) (mappedComm (f L) (comm L) (height L))
+  let p : E → X := mapPoint (f L) (comm L) (height L)
+  have hp : IsQuotientCoveringMap p (height L).ker :=
+    mapPoint_covering (f L) (comm L) (height L) (height L) (height_generator L)
+      (exponent L) (exponent_generator L) (height_surjective L)
+  have hs : Function.Surjective p := mapPoint_surjective (f L) (comm L) (height L) (height_surjective L)
+  let : PathConnectedSpace X := hs.pathConnectedSpace hp.continuous
+  let e : E := Classical.choice (inferInstance : Nonempty E)
+  refine ⟨X,inferInstance,inferInstance,inferInstance,?_,inferInstance,p e,?_,
+    E,inferInstance,inferInstance,p,hp.isCoveringMap,hs⟩
+  · intro k hk
+    exact ⟨fun c => Nat.not_lt_of_ge ((c.property.symm ▸ hd c.val.directions c.val.clique)) hk⟩
+  · exact ⟨(hp.fundamentalGroupEquiv ⟨e,rfl⟩).trans (MulEquiv.inv' ((height L).ker)).symm⟩
+
+end ArtinCubicalTopology
+
+/-- The fixed source group has an ordinary three-dimensional classifying CW
+space, with an actually contractible surjective covering. -/
+theorem source_hasClassifyingSpace_three : HasClassifyingSpace.{0} SourceGroup 3 :=
+  ArtinCubicalTopology.height_hasClassifyingSpace seedGraph 3 SeedConnection.clique_card_le_three
+end EilenbergGanea
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/Dimension.lean

@@ -1,115 +1,115 @@
-import OAI.Topology.EilenbergGanea.SeedContraction
-import OAI.Topology.EilenbergGanea.LevelBasis
-import OAI.Topology.EilenbergGanea.CliqueCohomology
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set Filter Topology MeasureTheory
-open scoped Quaternion ContDiff
-
-namespace EilenbergGanea.SeedEncoding
-@[simp] theorem cellEquiv_symm_apply (i : Fin 103) : cellEquiv.symm i = cellAt i := rfl
-end EilenbergGanea.SeedEncoding
-namespace EilenbergGanea
-noncomputable abbrev sourceVertexOrder : LinearOrder SeedCell :=
-  LinearOrder.lift' SeedEncoding.cellEquiv SeedEncoding.cellEquiv.injective
-local instance : LinearOrder SeedCell := sourceVertexOrder
-local instance : Preorder SeedCell := sourceVertexOrder.toPartialOrder.toPreorder
-local instance : LT SeedCell := sourceVertexOrder.toLT
-local instance : LE SeedCell := sourceVertexOrder.toLE
-
-@[simp] theorem sourceVertexOrder_lt (a b : SeedCell) :
-    a < b ↔ SeedEncoding.cellEquiv a < SeedEncoding.cellEquiv b := Iff.rfl
-
-theorem source_valid_to_index (w : List SeedCell) (hw : SimplicialChains.Valid seedGraph w) :
-    SimplicialChains.Valid SeedChainCertificate.graph (w.map SeedEncoding.cellEquiv) := by
-  constructor
-  · rw [List.pairwise_map]
-    exact hw.1
-  · rw [List.pairwise_map]
-    simpa only [SeedChainCertificate.graph,SimpleGraph.comap_adj,
-      ← SeedEncoding.cellEquiv_symm_apply,Equiv.symm_apply_apply] using hw.2
-
-theorem index_valid_to_source (w : List (Fin 103))
-    (hw : SimplicialChains.Valid SeedChainCertificate.graph w) :
-    SimplicialChains.Valid seedGraph (w.map SeedEncoding.cellEquiv.symm) := by
-  constructor
-  · rw [List.pairwise_map]
-    simpa only [sourceVertexOrder_lt,Equiv.apply_symm_apply] using hw.1
-  · rw [List.pairwise_map]
-    exact hw.2
-
-/-- Relabeling uses the original seed graph and merely chooses an order on its
-vertices for the signed integral chain convention. -/
-noncomputable def sourceLinkContraction (K : SimplicialChains.Contraction SeedChainCertificate.graph) :
-    SimplicialChains.Contraction seedGraph :=
-  K.reindex SeedEncoding.cellEquiv.symm (index_valid_to_source) (source_valid_to_index)
-
-end EilenbergGanea
-
-namespace EilenbergGanea
-local instance : LinearOrder SeedCell := sourceVertexOrder
-local instance : Preorder SeedCell := sourceVertexOrder.toPartialOrder.toPreorder
-local instance : LT SeedCell := sourceVertexOrder.toLT
-local instance : LE SeedCell := sourceVertexOrder.toLE
-
-theorem source_valid_length_le (w : List SeedCell) (hw : SimplicialChains.Valid seedGraph w) :
-    w.length ≤ 3 := by
-  simpa only [List.length_map] using
-    SeedChainCertificate.valid_length_le (w.map SeedEncoding.cellEquiv) (source_valid_to_index w hw)
-
-theorem source_cd_eq_two_of_contraction
-    (K : SimplicialChains.Contraction SeedChainCertificate.graph) :
-    cohomologicalDimension SourceGroup = 2 := by
-  apply le_antisymm
-  · exact LevelBasis.artin_height_kernel_cd_le_two seedGraph seedBase seed_connected.preconnected
-      (sourceLinkContraction K) source_valid_length_le
-  · exact source_two_le_cohomologicalDimension
-
-/-- The exact remaining geometric interface is not limited to finite or
-countable presentations. This endpoint excludes any ordinary presentation
-whose actual integral equivariant cellular boundaries freely span cycles. -/
-theorem source_no_boundary_basis_of_contraction
-    (K : SimplicialChains.Contraction SeedChainCertificate.graph)
-    {A R : Type*} (a₀ : A → SourceGroup)
-    (ha₀ : Function.Surjective (wordValue a₀))
-    (r₀ : R → FreeGroup A) (hr₀ : ∀ i, wordValue a₀ (r₀ i) = 1)
-    (hp₀ : (wordValue a₀).ker = Subgroup.normalClosure (Set.range r₀))
-    (E : BasedChains SourceGroup R ≃ₗ[ℤ] cycles a₀)
-    (hE : ∀ c, (E c : BasedChains SourceGroup A) = relatorChainMap a₀ r₀ c) : False := by
-  let : Group.ResiduallyFinite SourceGroup := source_residuallyFinite
-  exact LevelBasis.no_boundary_basis_presentation seedGraph seedBase seed_connected.preconnected
-    (sourceLinkContraction K) source_valid_length_le SourceTransport.source_small_labels
-    a₀ ha₀ r₀ hr₀ hp₀ E hE
-
-end EilenbergGanea
-
-
-
-namespace EilenbergGanea
-
-/-- Literal integral cohomological dimension of the source's fixed group. -/
-theorem source_cohomologicalDimension : cohomologicalDimension SourceGroup = 2 :=
-  source_cd_eq_two_of_contraction SeedChainCertificate.contraction
-
-/-- No arbitrary-alphabet genuine presentation has its relators as an integral
-basis of Cayley cycles.  -/
-theorem source_no_boundary_basis
-    {A R : Type*} (a₀ : A → SourceGroup)
-    (ha₀ : Function.Surjective (wordValue a₀))
-    (r₀ : R → FreeGroup A) (hr₀ : ∀ i, wordValue a₀ (r₀ i) = 1)
-    (hp₀ : (wordValue a₀).ker = Subgroup.normalClosure (Set.range r₀))
-    (E : BasedChains SourceGroup R ≃ₗ[ℤ] cycles a₀)
-    (hE : ∀ c, (E c : BasedChains SourceGroup A) = relatorChainMap a₀ r₀ c) : False :=
-  source_no_boundary_basis_of_contraction SeedChainCertificate.contraction
-    a₀ ha₀ r₀ hr₀ hp₀ E hE
-
-end EilenbergGanea
-
-
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.SeedContraction
+import OAI.Topology.EilenbergGanea.LevelBasis
+import OAI.Topology.EilenbergGanea.CliqueCohomology
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set Filter Topology MeasureTheory
+open scoped Quaternion ContDiff
+
+namespace EilenbergGanea.SeedEncoding
+@[simp] theorem cellEquiv_symm_apply (i : Fin 103) : cellEquiv.symm i = cellAt i := rfl
+end EilenbergGanea.SeedEncoding
+namespace EilenbergGanea
+noncomputable abbrev sourceVertexOrder : LinearOrder SeedCell :=
+  LinearOrder.lift' SeedEncoding.cellEquiv SeedEncoding.cellEquiv.injective
+local instance : LinearOrder SeedCell := sourceVertexOrder
+local instance : Preorder SeedCell := sourceVertexOrder.toPartialOrder.toPreorder
+local instance : LT SeedCell := sourceVertexOrder.toLT
+local instance : LE SeedCell := sourceVertexOrder.toLE
+
+@[simp] theorem sourceVertexOrder_lt (a b : SeedCell) :
+    a < b ↔ SeedEncoding.cellEquiv a < SeedEncoding.cellEquiv b := Iff.rfl
+
+theorem source_valid_to_index (w : List SeedCell) (hw : SimplicialChains.Valid seedGraph w) :
+    SimplicialChains.Valid SeedChainCertificate.graph (w.map SeedEncoding.cellEquiv) := by
+  constructor
+  · rw [List.pairwise_map]
+    exact hw.1
+  · rw [List.pairwise_map]
+    simpa only [SeedChainCertificate.graph,SimpleGraph.comap_adj,
+      ← SeedEncoding.cellEquiv_symm_apply,Equiv.symm_apply_apply] using hw.2
+
+theorem index_valid_to_source (w : List (Fin 103))
+    (hw : SimplicialChains.Valid SeedChainCertificate.graph w) :
+    SimplicialChains.Valid seedGraph (w.map SeedEncoding.cellEquiv.symm) := by
+  constructor
+  · rw [List.pairwise_map]
+    simpa only [sourceVertexOrder_lt,Equiv.apply_symm_apply] using hw.1
+  · rw [List.pairwise_map]
+    exact hw.2
+
+/-- Relabeling uses the original seed graph and merely chooses an order on its
+vertices for the signed integral chain convention. -/
+noncomputable def sourceLinkContraction (K : SimplicialChains.Contraction SeedChainCertificate.graph) :
+    SimplicialChains.Contraction seedGraph :=
+  K.reindex SeedEncoding.cellEquiv.symm (index_valid_to_source) (source_valid_to_index)
+
+end EilenbergGanea
+
+namespace EilenbergGanea
+local instance : LinearOrder SeedCell := sourceVertexOrder
+local instance : Preorder SeedCell := sourceVertexOrder.toPartialOrder.toPreorder
+local instance : LT SeedCell := sourceVertexOrder.toLT
+local instance : LE SeedCell := sourceVertexOrder.toLE
+
+theorem source_valid_length_le (w : List SeedCell) (hw : SimplicialChains.Valid seedGraph w) :
+    w.length ≤ 3 := by
+  simpa only [List.length_map] using
+    SeedChainCertificate.valid_length_le (w.map SeedEncoding.cellEquiv) (source_valid_to_index w hw)
+
+theorem source_cd_eq_two_of_contraction
+    (K : SimplicialChains.Contraction SeedChainCertificate.graph) :
+    cohomologicalDimension SourceGroup = 2 := by
+  apply le_antisymm
+  · exact LevelBasis.artin_height_kernel_cd_le_two seedGraph seedBase seed_connected.preconnected
+      (sourceLinkContraction K) source_valid_length_le
+  · exact source_two_le_cohomologicalDimension
+
+/-- The exact remaining geometric interface is not limited to finite or
+countable presentations. This endpoint excludes any ordinary presentation
+whose actual integral equivariant cellular boundaries freely span cycles. -/
+theorem source_no_boundary_basis_of_contraction
+    (K : SimplicialChains.Contraction SeedChainCertificate.graph)
+    {A R : Type*} (a₀ : A → SourceGroup)
+    (ha₀ : Function.Surjective (wordValue a₀))
+    (r₀ : R → FreeGroup A) (hr₀ : ∀ i, wordValue a₀ (r₀ i) = 1)
+    (hp₀ : (wordValue a₀).ker = Subgroup.normalClosure (Set.range r₀))
+    (E : BasedChains SourceGroup R ≃ₗ[ℤ] cycles a₀)
+    (hE : ∀ c, (E c : BasedChains SourceGroup A) = relatorChainMap a₀ r₀ c) : False := by
+  let : Group.ResiduallyFinite SourceGroup := source_residuallyFinite
+  exact LevelBasis.no_boundary_basis_presentation seedGraph seedBase seed_connected.preconnected
+    (sourceLinkContraction K) source_valid_length_le SourceTransport.source_small_labels
+    a₀ ha₀ r₀ hr₀ hp₀ E hE
+
+end EilenbergGanea
+
+
+
+namespace EilenbergGanea
+
+/-- Literal integral cohomological dimension of the source's fixed group. -/
+theorem source_cohomologicalDimension : cohomologicalDimension SourceGroup = 2 :=
+  source_cd_eq_two_of_contraction SeedChainCertificate.contraction
+
+/-- No arbitrary-alphabet genuine presentation has its relators as an integral
+basis of Cayley cycles.  -/
+theorem source_no_boundary_basis
+    {A R : Type*} (a₀ : A → SourceGroup)
+    (ha₀ : Function.Surjective (wordValue a₀))
+    (r₀ : R → FreeGroup A) (hr₀ : ∀ i, wordValue a₀ (r₀ i) = 1)
+    (hp₀ : (wordValue a₀).ker = Subgroup.normalClosure (Set.range r₀))
+    (E : BasedChains SourceGroup R ≃ₗ[ℤ] cycles a₀)
+    (hE : ∀ c, (E c : BasedChains SourceGroup A) = relatorChainMap a₀ r₀ c) : False :=
+  source_no_boundary_basis_of_contraction SeedChainCertificate.contraction
+    a₀ ha₀ r₀ hr₀ hp₀ E hE
+
+end EilenbergGanea
+
+
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/IntegralModules.lean

@@ -1,308 +1,308 @@
-import OAI.Topology.EilenbergGanea.Model
-import OAI.Topology.EilenbergGanea.PresentationAlignment
-import Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension
-import Mathlib.RingTheory.Noetherian.Basic
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set
-
-/-! Group-ring augmentation and literal two-syzygy dimension criterion. -/
-namespace EilenbergGanea
-open CategoryTheory
-
-section ProjectiveBound
-variable {R P Q M : Type} [Ring R]
-  [AddCommGroup P] [Module R P] [AddCommGroup Q] [Module R Q]
-  [AddCommGroup M] [Module R M]
-
-/-- Two successive projective syzygies give the literal integral upper bound. -/
-theorem projectiveDimension_le_two_of_syzygies
-    (ε : P →ₗ[R] M) (hε : Function.Surjective ε)
-    (d : Q →ₗ[R] LinearMap.ker ε) (hd : Function.Surjective d)
-    [Module.Projective R P] [Module.Projective R Q]
-    [Module.Projective R (LinearMap.ker d)] :
-    CategoryTheory.projectiveDimension (ModuleCat.of R M) ≤ 2 := by
-  have h₁ := (LinearMap.shortExact_shortComplexKer hd).hasProjectiveDimensionLT_X₃ 1
-    (inferInstance : HasProjectiveDimensionLT (ModuleCat.of R (LinearMap.ker d)) 1)
-    (inferInstance : HasProjectiveDimensionLT (ModuleCat.of R Q) 2)
-  have h₂ := (LinearMap.shortExact_shortComplexKer hε).hasProjectiveDimensionLT_X₃ 2
-    h₁ (inferInstance : HasProjectiveDimensionLT (ModuleCat.of R P) 3)
-  exact (CategoryTheory.projectiveDimension_le_iff _ 2).mpr h₂
-end ProjectiveBound
-
-section CayleyModules
-variable {Γ S : Type} [Group Γ]
-
-abbrev IntegralGroupRing (Γ : Type) [Group Γ] := MonoidAlgebra ℤ Γ
-abbrev IntegralTrivialModule (Γ : Type) [Group Γ] :=
-  (Representation.trivial ℤ Γ ℤ).asModule
-
-def cayleyAugmentation : IntegralGroupRing Γ →ₗ[IntegralGroupRing Γ] IntegralTrivialModule Γ :=
-  LinearMap.toSpanSingleton (IntegralGroupRing Γ) _
-    ((Representation.trivial ℤ Γ ℤ).asModuleEquiv.symm 1)
-
-@[simp] theorem cayleyAugmentation_single (g : Γ) (n : ℤ) :
-    cayleyAugmentation (MonoidAlgebra.single g n) =
-      (Representation.trivial ℤ Γ ℤ).asModuleEquiv.symm n := by
-  change (MonoidAlgebra.single g n : IntegralGroupRing Γ) •
-    ((Representation.trivial ℤ Γ ℤ).asModuleEquiv.symm 1) = _
-  rw [Representation.single_smul]
-  change (n : ℤ) * 1 = n
-  exact mul_one n
-
-theorem cayleyAugmentation_surjective :
-    Function.Surjective (cayleyAugmentation (Γ := Γ)) := by
-  intro n
-  refine ⟨MonoidAlgebra.single (1 : Γ) ((Representation.trivial ℤ Γ ℤ).asModuleEquiv n),?_⟩
-  rw [cayleyAugmentation_single,LinearEquiv.symm_apply_apply]
-
-def cayleyDifferential (v : S → Γ) :
-    (S →₀ IntegralGroupRing Γ) →ₗ[IntegralGroupRing Γ] IntegralGroupRing Γ :=
-  Finsupp.linearCombination _ (fun s => MonoidAlgebra.single (v s) 1 - 1)
-
-@[simp] theorem cayleyDifferential_single (v : S → Γ) (s : S) (r : IntegralGroupRing Γ) :
-    cayleyDifferential v (Finsupp.single s r) = r * (MonoidAlgebra.single (v s) 1 - 1) := by
-  simp [cayleyDifferential]
-
-@[simp] theorem cayleyAugmentation_one :
-    cayleyAugmentation (1 : IntegralGroupRing Γ) =
-      (Representation.trivial ℤ Γ ℤ).asModuleEquiv.symm 1 :=
-  cayleyAugmentation_single (1 : Γ) 1
-
-@[simp] theorem cayleyAugmentation_difference (g : Γ) :
-    cayleyAugmentation (MonoidAlgebra.single g 1 - 1) = 0 := by
-  rw [map_sub, cayleyAugmentation_single, cayleyAugmentation_one, sub_self]
-
-theorem cayleyDifferential_aug (v : S → Γ) :
-    (cayleyAugmentation (Γ := Γ)).comp (cayleyDifferential v) = 0 := by
-  change (cayleyAugmentation (Γ := Γ)).comp
-    (Finsupp.linearCombination _ (fun s => MonoidAlgebra.single (v s) 1 - 1)) = 0
-  rw [← Finsupp.linearCombination_linear_comp]
-  have he : (cayleyAugmentation (Γ := Γ)) ∘
-      (fun s => MonoidAlgebra.single (v s) 1 - 1) = (0 : S → IntegralTrivialModule Γ) := by
-    funext s
-    exact cayleyAugmentation_difference (v s)
-  rw [he, Finsupp.linearCombination_zero]
-
- theorem cayleyGeneratorDifference_mem (v : S → Γ) (w : FreeGroup S) :
-    MonoidAlgebra.single (FreeGroup.lift v w) 1 - 1 ∈ LinearMap.range (cayleyDifferential v) := by
-  induction w using FreeGroup.induction_on with
-  | one =>
-      simp only [map_one]
-      have he : (MonoidAlgebra.single (1 : Γ) 1 : IntegralGroupRing Γ) = 1 := rfl
-      rw [he,sub_self]
-      exact Submodule.zero_mem _
-  | of s =>
-      exact ⟨Finsupp.single s 1,by simp⟩
-  | inv_of s =>
-      refine ⟨Finsupp.single s (-MonoidAlgebra.single (v s)⁻¹ 1),?_⟩
-      simp [mul_sub,MonoidAlgebra.single_mul_single]
-      rfl
-  | mul w z hw hz =>
-      have hh := (LinearMap.range (cayleyDifferential v)).smul_mem
-        (MonoidAlgebra.single (FreeGroup.lift v w) 1) hz
-      have hc := (LinearMap.range (cayleyDifferential v)).add_mem hw hh
-      rw [map_mul]
-      convert hc using 1
-      change _ = (_ - 1) + _ * (_ - 1)
-      rw [mul_sub,mul_one,MonoidAlgebra.single_mul_single,one_mul]
-      abel
-
-theorem cayleyAugmentation_normalize (v : S → Γ)
-    (hv : Function.Surjective (FreeGroup.lift v)) (r : IntegralGroupRing Γ) :
-    r - MonoidAlgebra.single 1
-      ((Representation.trivial ℤ Γ ℤ).asModuleEquiv (cayleyAugmentation r)) ∈
-        LinearMap.range (cayleyDifferential v) := by
-  classical
-  induction r using MonoidAlgebra.induction_linear with
-  | zero => simp
-  | add r t hr ht =>
-      rw [map_add, map_add, MonoidAlgebra.single_add]
-      convert (LinearMap.range (cayleyDifferential v)).add_mem hr ht using 1; abel
-  | single g n =>
-      rw [cayleyAugmentation_single, LinearEquiv.apply_symm_apply]
-      obtain ⟨w, hw⟩ := hv g
-      have hm := (LinearMap.range (cayleyDifferential v)).toAddSubgroup.zsmul_mem
-        (cayleyGeneratorDifference_mem v w) n
-      rw [hw] at hm
-      convert hm using 1
-      simp only [smul_sub, MonoidAlgebra.smul_single, zsmul_eq_mul, mul_one]
-      rfl
-
-theorem cayley_exact (v : S → Γ) (hv : Function.Surjective (FreeGroup.lift v)) :
-    LinearMap.range (cayleyDifferential v) = LinearMap.ker (cayleyAugmentation (Γ := Γ)) := by
-  apply le_antisymm
-  · rintro _ ⟨r,rfl⟩
-    exact LinearMap.congr_fun (cayleyDifferential_aug v) r
-  · intro r hr
-    have hm := cayleyAugmentation_normalize v hv r
-    rw [LinearMap.mem_ker.mp hr, map_zero, MonoidAlgebra.single_zero, sub_zero] at hm
-    exact hm
-
-def cayleySyzygyDifferential (v : S → Γ) :
-    (S →₀ IntegralGroupRing Γ) →ₗ[IntegralGroupRing Γ]
-      LinearMap.ker (cayleyAugmentation (Γ := Γ)) :=
-  (cayleyDifferential v).codRestrict _ (by
-    intro r
-    exact LinearMap.congr_fun (cayleyDifferential_aug v) r)
-
-theorem cayleySyzygyDifferential_surjective (v : S → Γ)
-    (hv : Function.Surjective (FreeGroup.lift v)) :
-    Function.Surjective (cayleySyzygyDifferential v) := by
-  intro r
-  have hr : (r : IntegralGroupRing Γ) ∈ LinearMap.range (cayleyDifferential v) := by
-    rw [cayley_exact v hv]
-    exact r.property
-  obtain ⟨s, hs⟩ := hr
-  exact ⟨s, Subtype.ext hs⟩
-
-theorem cayley_projectiveDimension_le_two (v : S → Γ)
-    (hv : Function.Surjective (FreeGroup.lift v))
-    [Module.Projective (IntegralGroupRing Γ) (LinearMap.ker (cayleyDifferential v))] :
-    CategoryTheory.projectiveDimension
-      (ModuleCat.of (IntegralGroupRing Γ) (IntegralTrivialModule Γ)) ≤ 2 := by
-  have hk : LinearMap.ker (cayleySyzygyDifferential v) =
-      LinearMap.ker (cayleyDifferential v) := LinearMap.ker_codRestrict _ _ _
-  have : Module.Projective (IntegralGroupRing Γ)
-      (LinearMap.ker (cayleySyzygyDifferential v)) := hk ▸ inferInstance
-  exact projectiveDimension_le_two_of_syzygies cayleyAugmentation
-    cayleyAugmentation_surjective (cayleySyzygyDifferential v)
-    (cayleySyzygyDifferential_surjective v hv)
-
-end CayleyModules
-end EilenbergGanea
-
-namespace EilenbergGanea
-section ChainPackaging
-variable {Γ S T : Type} [Group Γ]
-
-/-- Reindex the actual integral edge chains as a free left group-ring module. -/
-def chainPackaging : BasedChains Γ S ≃ₗ[ℤ] S →₀ IntegralGroupRing Γ :=
-  ((Finsupp.domLCongr (Equiv.prodComm Γ S)).trans (Finsupp.curryLinearEquiv ℤ)).trans
-    (Finsupp.mapRange.linearEquiv (MonoidAlgebra.coeffLinearEquiv ℤ).symm)
-
-@[simp] theorem chainPackaging_single (g : Γ) (s : S) (n : ℤ) :
-    chainPackaging (Finsupp.single (g,s) n) =
-      Finsupp.single s (MonoidAlgebra.single g n) := by
-  simp [chainPackaging, Finsupp.curry_single]
-
-@[simp] theorem chainPackaging_symm_single (g : Γ) (s : S) (n : ℤ) :
-    (chainPackaging (Γ := Γ) (S := S)).symm
-        (Finsupp.single s (MonoidAlgebra.single g n)) = Finsupp.single (g,s) n := by
-  apply (chainPackaging (Γ := Γ) (S := S)).injective
-  simp
-
-theorem chainPackaging_translate (g : Γ) (c : BasedChains Γ S) :
-    chainPackaging (translateChains g c) =
-      (MonoidAlgebra.single g 1 : IntegralGroupRing Γ) • chainPackaging c := by
-  have he : (chainPackaging (Γ := Γ) (S := S)).toLinearMap.comp (translateChains g) =
-      (DistribSMul.toLinearMap ℤ (S →₀ IntegralGroupRing Γ) (MonoidAlgebra.single g 1 : IntegralGroupRing Γ)).comp
-        (chainPackaging (Γ := Γ) (S := S)).toLinearMap := by
-    apply Finsupp.lhom_ext
-    rintro ⟨h,s⟩ n
-    simp [Finsupp.smul_single, MonoidAlgebra.single_mul_single]
-  exact LinearMap.congr_fun he c
-
-theorem chainPackaging_boundary (v : S → Γ) (c : BasedChains Γ S) :
-    cayleyDifferential v (chainPackaging c) = MonoidAlgebra.ofCoeff (edgeBoundary v c) := by
-  have he : ((cayleyDifferential v).restrictScalars ℤ).comp
-      (chainPackaging (Γ := Γ) (S := S)).toLinearMap =
-      (MonoidAlgebra.coeffLinearEquiv ℤ).symm.toLinearMap.comp (edgeBoundary v) := by
-    apply Finsupp.lhom_ext
-    rintro ⟨g,s⟩ n
-    simp only [LinearMap.comp_apply, LinearMap.coe_restrictScalars, LinearEquiv.coe_coe,
-      chainPackaging_single, cayleyDifferential_single, edgeBoundary_single, map_smul,
-      map_sub]
-    simp only [mul_sub, MonoidAlgebra.single_mul_single, mul_one,
-      MonoidAlgebra.coeffLinearEquiv_symm_apply, MonoidAlgebra.ofCoeff_single,
-      MonoidAlgebra.smul_single, smul_sub, smul_eq_mul]
-  exact LinearMap.congr_fun he c
-
-theorem chainPackaging_symm_translate (g : Γ) (c : S →₀ IntegralGroupRing Γ) :
-    (chainPackaging (Γ := Γ) (S := S)).symm ((MonoidAlgebra.single g 1 : IntegralGroupRing Γ) • c) =
-      translateChains g (chainPackaging.symm c) := by
-  apply (chainPackaging (Γ := Γ) (S := S)).injective
-  rw [chainPackaging_translate, LinearEquiv.apply_symm_apply, LinearEquiv.apply_symm_apply]
-
-/-- Equivariance on integral chains is exactly full group-ring linearity. -/
-def chainRingMap (A : BasedChains Γ S →ₗ[ℤ] BasedChains Γ T)
-    (hA : ChainsEquivariant A) :
-    (S →₀ IntegralGroupRing Γ) →ₗ[IntegralGroupRing Γ] T →₀ IntegralGroupRing Γ where
-  toFun c := chainPackaging (A (chainPackaging.symm c))
-  map_add' _ _ := by simp only [map_add]
-  map_smul' r c := by
-    induction r using MonoidAlgebra.induction_linear with
-    | zero => simp only [zero_smul, map_zero]
-    | add r t hr ht => simp only [add_smul, map_add, hr, ht]
-    | single g n =>
-        have hn : (MonoidAlgebra.single g n : IntegralGroupRing Γ) =
-            n • MonoidAlgebra.single g 1 := by simp
-        rw [hn]
-        simp only [smul_assoc, map_smul, chainPackaging_symm_translate, RingHom.id_apply]
-        rw [hA g _, chainPackaging_translate]
-
-@[simp] theorem chainRingMap_apply (A : BasedChains Γ S →ₗ[ℤ] BasedChains Γ T)
-    (hA : ChainsEquivariant A) (c : S →₀ IntegralGroupRing Γ) :
-    chainRingMap A hA c = chainPackaging (A (chainPackaging.symm c)) := rfl
-
-theorem chainPackaging_mem_cycles (v : S → Γ) (c : BasedChains Γ S) :
-    c ∈ cycles v ↔ chainPackaging c ∈ LinearMap.ker (cayleyDifferential v) := by
-  change edgeBoundary v c = 0 ↔ cayleyDifferential v (chainPackaging c) = 0
-  rw [chainPackaging_boundary]
-  constructor
-  · intro h
-    rw [h]
-    rfl
-  · intro h
-    have hh := congrArg MonoidAlgebra.coeff h
-    exact hh
-
-theorem projective_cayley_cycles_of_boundary_basis (v : S → Γ)
-    (A : BasedChains Γ T →ₗ[ℤ] BasedChains Γ S) (hA : ChainsEquivariant A)
-    (hAi : Function.Injective A) (hAr : LinearMap.range A = cycles v) :
-    Module.Projective (IntegralGroupRing Γ) (LinearMap.ker (cayleyDifferential v)) := by
-  have hm : ∀ c, chainRingMap A hA c ∈ LinearMap.ker (cayleyDifferential v) := by
-    intro c
-    apply (chainPackaging_mem_cycles v _).mp
-    rw [← hAr]
-    exact ⟨chainPackaging.symm c,rfl⟩
-  let d := (chainRingMap A hA).codRestrict _ hm
-  have hi : Function.Injective d := by
-    intro c c' h
-    apply (chainPackaging (Γ := Γ) (S := T)).symm.injective
-    apply hAi
-    apply chainPackaging.injective
-    exact congrArg Subtype.val h
-  have hs : Function.Surjective d := by
-    intro c
-    have hc : chainPackaging.symm (c : S →₀ IntegralGroupRing Γ) ∈ cycles v := by
-      apply (chainPackaging_mem_cycles v _).mpr
-      rw [LinearEquiv.apply_symm_apply]
-      exact c.property
-    rw [← hAr] at hc
-    obtain ⟨b,hb⟩ := hc
-    refine ⟨chainPackaging b,Subtype.ext ?_⟩
-    change chainPackaging (A (chainPackaging.symm (chainPackaging b))) = c.val
-    rw [LinearEquiv.symm_apply_apply,hb,LinearEquiv.apply_symm_apply]
-  exact Module.Projective.of_equiv' (LinearEquiv.ofBijective d ⟨hi,hs⟩)
-
-theorem projectiveDimension_le_two_of_boundary_basis (v : S → Γ)
-    (hv : Function.Surjective (FreeGroup.lift v))
-    (A : BasedChains Γ T →ₗ[ℤ] BasedChains Γ S) (hA : ChainsEquivariant A)
-    (hAi : Function.Injective A) (hAr : LinearMap.range A = cycles v) :
-    CategoryTheory.projectiveDimension
-      (ModuleCat.of (IntegralGroupRing Γ) (IntegralTrivialModule Γ)) ≤ 2 := by
-  have := projective_cayley_cycles_of_boundary_basis v A hA hAi hAr
-  exact cayley_projectiveDimension_le_two v hv
-
-end ChainPackaging
-
-
-end EilenbergGanea
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.Model
+import OAI.Topology.EilenbergGanea.PresentationAlignment
+import Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension
+import Mathlib.RingTheory.Noetherian.Basic
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set
+
+/-! Group-ring augmentation and literal two-syzygy dimension criterion. -/
+namespace EilenbergGanea
+open CategoryTheory
+
+section ProjectiveBound
+variable {R P Q M : Type} [Ring R]
+  [AddCommGroup P] [Module R P] [AddCommGroup Q] [Module R Q]
+  [AddCommGroup M] [Module R M]
+
+/-- Two successive projective syzygies give the literal integral upper bound. -/
+theorem projectiveDimension_le_two_of_syzygies
+    (ε : P →ₗ[R] M) (hε : Function.Surjective ε)
+    (d : Q →ₗ[R] LinearMap.ker ε) (hd : Function.Surjective d)
+    [Module.Projective R P] [Module.Projective R Q]
+    [Module.Projective R (LinearMap.ker d)] :
+    CategoryTheory.projectiveDimension (ModuleCat.of R M) ≤ 2 := by
+  have h₁ := (LinearMap.shortExact_shortComplexKer hd).hasProjectiveDimensionLT_X₃ 1
+    (inferInstance : HasProjectiveDimensionLT (ModuleCat.of R (LinearMap.ker d)) 1)
+    (inferInstance : HasProjectiveDimensionLT (ModuleCat.of R Q) 2)
+  have h₂ := (LinearMap.shortExact_shortComplexKer hε).hasProjectiveDimensionLT_X₃ 2
+    h₁ (inferInstance : HasProjectiveDimensionLT (ModuleCat.of R P) 3)
+  exact (CategoryTheory.projectiveDimension_le_iff _ 2).mpr h₂
+end ProjectiveBound
+
+section CayleyModules
+variable {Γ S : Type} [Group Γ]
+
+abbrev IntegralGroupRing (Γ : Type) [Group Γ] := MonoidAlgebra ℤ Γ
+abbrev IntegralTrivialModule (Γ : Type) [Group Γ] :=
+  (Representation.trivial ℤ Γ ℤ).asModule
+
+def cayleyAugmentation : IntegralGroupRing Γ →ₗ[IntegralGroupRing Γ] IntegralTrivialModule Γ :=
+  LinearMap.toSpanSingleton (IntegralGroupRing Γ) _
+    ((Representation.trivial ℤ Γ ℤ).asModuleEquiv.symm 1)
+
+@[simp] theorem cayleyAugmentation_single (g : Γ) (n : ℤ) :
+    cayleyAugmentation (MonoidAlgebra.single g n) =
+      (Representation.trivial ℤ Γ ℤ).asModuleEquiv.symm n := by
+  change (MonoidAlgebra.single g n : IntegralGroupRing Γ) •
+    ((Representation.trivial ℤ Γ ℤ).asModuleEquiv.symm 1) = _
+  rw [Representation.single_smul]
+  change (n : ℤ) * 1 = n
+  exact mul_one n
+
+theorem cayleyAugmentation_surjective :
+    Function.Surjective (cayleyAugmentation (Γ := Γ)) := by
+  intro n
+  refine ⟨MonoidAlgebra.single (1 : Γ) ((Representation.trivial ℤ Γ ℤ).asModuleEquiv n),?_⟩
+  rw [cayleyAugmentation_single,LinearEquiv.symm_apply_apply]
+
+def cayleyDifferential (v : S → Γ) :
+    (S →₀ IntegralGroupRing Γ) →ₗ[IntegralGroupRing Γ] IntegralGroupRing Γ :=
+  Finsupp.linearCombination _ (fun s => MonoidAlgebra.single (v s) 1 - 1)
+
+@[simp] theorem cayleyDifferential_single (v : S → Γ) (s : S) (r : IntegralGroupRing Γ) :
+    cayleyDifferential v (Finsupp.single s r) = r * (MonoidAlgebra.single (v s) 1 - 1) := by
+  simp [cayleyDifferential]
+
+@[simp] theorem cayleyAugmentation_one :
+    cayleyAugmentation (1 : IntegralGroupRing Γ) =
+      (Representation.trivial ℤ Γ ℤ).asModuleEquiv.symm 1 :=
+  cayleyAugmentation_single (1 : Γ) 1
+
+@[simp] theorem cayleyAugmentation_difference (g : Γ) :
+    cayleyAugmentation (MonoidAlgebra.single g 1 - 1) = 0 := by
+  rw [map_sub, cayleyAugmentation_single, cayleyAugmentation_one, sub_self]
+
+theorem cayleyDifferential_aug (v : S → Γ) :
+    (cayleyAugmentation (Γ := Γ)).comp (cayleyDifferential v) = 0 := by
+  change (cayleyAugmentation (Γ := Γ)).comp
+    (Finsupp.linearCombination _ (fun s => MonoidAlgebra.single (v s) 1 - 1)) = 0
+  rw [← Finsupp.linearCombination_linear_comp]
+  have he : (cayleyAugmentation (Γ := Γ)) ∘
+      (fun s => MonoidAlgebra.single (v s) 1 - 1) = (0 : S → IntegralTrivialModule Γ) := by
+    funext s
+    exact cayleyAugmentation_difference (v s)
+  rw [he, Finsupp.linearCombination_zero]
+
+ theorem cayleyGeneratorDifference_mem (v : S → Γ) (w : FreeGroup S) :
+    MonoidAlgebra.single (FreeGroup.lift v w) 1 - 1 ∈ LinearMap.range (cayleyDifferential v) := by
+  induction w using FreeGroup.induction_on with
+  | one =>
+      simp only [map_one]
+      have he : (MonoidAlgebra.single (1 : Γ) 1 : IntegralGroupRing Γ) = 1 := rfl
+      rw [he,sub_self]
+      exact Submodule.zero_mem _
+  | of s =>
+      exact ⟨Finsupp.single s 1,by simp⟩
+  | inv_of s =>
+      refine ⟨Finsupp.single s (-MonoidAlgebra.single (v s)⁻¹ 1),?_⟩
+      simp [mul_sub,MonoidAlgebra.single_mul_single]
+      rfl
+  | mul w z hw hz =>
+      have hh := (LinearMap.range (cayleyDifferential v)).smul_mem
+        (MonoidAlgebra.single (FreeGroup.lift v w) 1) hz
+      have hc := (LinearMap.range (cayleyDifferential v)).add_mem hw hh
+      rw [map_mul]
+      convert hc using 1
+      change _ = (_ - 1) + _ * (_ - 1)
+      rw [mul_sub,mul_one,MonoidAlgebra.single_mul_single,one_mul]
+      abel
+
+theorem cayleyAugmentation_normalize (v : S → Γ)
+    (hv : Function.Surjective (FreeGroup.lift v)) (r : IntegralGroupRing Γ) :
+    r - MonoidAlgebra.single 1
+      ((Representation.trivial ℤ Γ ℤ).asModuleEquiv (cayleyAugmentation r)) ∈
+        LinearMap.range (cayleyDifferential v) := by
+  classical
+  induction r using MonoidAlgebra.induction_linear with
+  | zero => simp
+  | add r t hr ht =>
+      rw [map_add, map_add, MonoidAlgebra.single_add]
+      convert (LinearMap.range (cayleyDifferential v)).add_mem hr ht using 1; abel
+  | single g n =>
+      rw [cayleyAugmentation_single, LinearEquiv.apply_symm_apply]
+      obtain ⟨w, hw⟩ := hv g
+      have hm := (LinearMap.range (cayleyDifferential v)).toAddSubgroup.zsmul_mem
+        (cayleyGeneratorDifference_mem v w) n
+      rw [hw] at hm
+      convert hm using 1
+      simp only [smul_sub, MonoidAlgebra.smul_single, zsmul_eq_mul, mul_one]
+      rfl
+
+theorem cayley_exact (v : S → Γ) (hv : Function.Surjective (FreeGroup.lift v)) :
+    LinearMap.range (cayleyDifferential v) = LinearMap.ker (cayleyAugmentation (Γ := Γ)) := by
+  apply le_antisymm
+  · rintro _ ⟨r,rfl⟩
+    exact LinearMap.congr_fun (cayleyDifferential_aug v) r
+  · intro r hr
+    have hm := cayleyAugmentation_normalize v hv r
+    rw [LinearMap.mem_ker.mp hr, map_zero, MonoidAlgebra.single_zero, sub_zero] at hm
+    exact hm
+
+def cayleySyzygyDifferential (v : S → Γ) :
+    (S →₀ IntegralGroupRing Γ) →ₗ[IntegralGroupRing Γ]
+      LinearMap.ker (cayleyAugmentation (Γ := Γ)) :=
+  (cayleyDifferential v).codRestrict _ (by
+    intro r
+    exact LinearMap.congr_fun (cayleyDifferential_aug v) r)
+
+theorem cayleySyzygyDifferential_surjective (v : S → Γ)
+    (hv : Function.Surjective (FreeGroup.lift v)) :
+    Function.Surjective (cayleySyzygyDifferential v) := by
+  intro r
+  have hr : (r : IntegralGroupRing Γ) ∈ LinearMap.range (cayleyDifferential v) := by
+    rw [cayley_exact v hv]
+    exact r.property
+  obtain ⟨s, hs⟩ := hr
+  exact ⟨s, Subtype.ext hs⟩
+
+theorem cayley_projectiveDimension_le_two (v : S → Γ)
+    (hv : Function.Surjective (FreeGroup.lift v))
+    [Module.Projective (IntegralGroupRing Γ) (LinearMap.ker (cayleyDifferential v))] :
+    CategoryTheory.projectiveDimension
+      (ModuleCat.of (IntegralGroupRing Γ) (IntegralTrivialModule Γ)) ≤ 2 := by
+  have hk : LinearMap.ker (cayleySyzygyDifferential v) =
+      LinearMap.ker (cayleyDifferential v) := LinearMap.ker_codRestrict _ _ _
+  have : Module.Projective (IntegralGroupRing Γ)
+      (LinearMap.ker (cayleySyzygyDifferential v)) := hk ▸ inferInstance
+  exact projectiveDimension_le_two_of_syzygies cayleyAugmentation
+    cayleyAugmentation_surjective (cayleySyzygyDifferential v)
+    (cayleySyzygyDifferential_surjective v hv)
+
+end CayleyModules
+end EilenbergGanea
+
+namespace EilenbergGanea
+section ChainPackaging
+variable {Γ S T : Type} [Group Γ]
+
+/-- Reindex the actual integral edge chains as a free left group-ring module. -/
+def chainPackaging : BasedChains Γ S ≃ₗ[ℤ] S →₀ IntegralGroupRing Γ :=
+  ((Finsupp.domLCongr (Equiv.prodComm Γ S)).trans (Finsupp.curryLinearEquiv ℤ)).trans
+    (Finsupp.mapRange.linearEquiv (MonoidAlgebra.coeffLinearEquiv ℤ).symm)
+
+@[simp] theorem chainPackaging_single (g : Γ) (s : S) (n : ℤ) :
+    chainPackaging (Finsupp.single (g,s) n) =
+      Finsupp.single s (MonoidAlgebra.single g n) := by
+  simp [chainPackaging, Finsupp.curry_single]
+
+@[simp] theorem chainPackaging_symm_single (g : Γ) (s : S) (n : ℤ) :
+    (chainPackaging (Γ := Γ) (S := S)).symm
+        (Finsupp.single s (MonoidAlgebra.single g n)) = Finsupp.single (g,s) n := by
+  apply (chainPackaging (Γ := Γ) (S := S)).injective
+  simp
+
+theorem chainPackaging_translate (g : Γ) (c : BasedChains Γ S) :
+    chainPackaging (translateChains g c) =
+      (MonoidAlgebra.single g 1 : IntegralGroupRing Γ) • chainPackaging c := by
+  have he : (chainPackaging (Γ := Γ) (S := S)).toLinearMap.comp (translateChains g) =
+      (DistribSMul.toLinearMap ℤ (S →₀ IntegralGroupRing Γ) (MonoidAlgebra.single g 1 : IntegralGroupRing Γ)).comp
+        (chainPackaging (Γ := Γ) (S := S)).toLinearMap := by
+    apply Finsupp.lhom_ext
+    rintro ⟨h,s⟩ n
+    simp [Finsupp.smul_single, MonoidAlgebra.single_mul_single]
+  exact LinearMap.congr_fun he c
+
+theorem chainPackaging_boundary (v : S → Γ) (c : BasedChains Γ S) :
+    cayleyDifferential v (chainPackaging c) = MonoidAlgebra.ofCoeff (edgeBoundary v c) := by
+  have he : ((cayleyDifferential v).restrictScalars ℤ).comp
+      (chainPackaging (Γ := Γ) (S := S)).toLinearMap =
+      (MonoidAlgebra.coeffLinearEquiv ℤ).symm.toLinearMap.comp (edgeBoundary v) := by
+    apply Finsupp.lhom_ext
+    rintro ⟨g,s⟩ n
+    simp only [LinearMap.comp_apply, LinearMap.coe_restrictScalars, LinearEquiv.coe_coe,
+      chainPackaging_single, cayleyDifferential_single, edgeBoundary_single, map_smul,
+      map_sub]
+    simp only [mul_sub, MonoidAlgebra.single_mul_single, mul_one,
+      MonoidAlgebra.coeffLinearEquiv_symm_apply, MonoidAlgebra.ofCoeff_single,
+      MonoidAlgebra.smul_single, smul_sub, smul_eq_mul]
+  exact LinearMap.congr_fun he c
+
+theorem chainPackaging_symm_translate (g : Γ) (c : S →₀ IntegralGroupRing Γ) :
+    (chainPackaging (Γ := Γ) (S := S)).symm ((MonoidAlgebra.single g 1 : IntegralGroupRing Γ) • c) =
+      translateChains g (chainPackaging.symm c) := by
+  apply (chainPackaging (Γ := Γ) (S := S)).injective
+  rw [chainPackaging_translate, LinearEquiv.apply_symm_apply, LinearEquiv.apply_symm_apply]
+
+/-- Equivariance on integral chains is exactly full group-ring linearity. -/
+def chainRingMap (A : BasedChains Γ S →ₗ[ℤ] BasedChains Γ T)
+    (hA : ChainsEquivariant A) :
+    (S →₀ IntegralGroupRing Γ) →ₗ[IntegralGroupRing Γ] T →₀ IntegralGroupRing Γ where
+  toFun c := chainPackaging (A (chainPackaging.symm c))
+  map_add' _ _ := by simp only [map_add]
+  map_smul' r c := by
+    induction r using MonoidAlgebra.induction_linear with
+    | zero => simp only [zero_smul, map_zero]
+    | add r t hr ht => simp only [add_smul, map_add, hr, ht]
+    | single g n =>
+        have hn : (MonoidAlgebra.single g n : IntegralGroupRing Γ) =
+            n • MonoidAlgebra.single g 1 := by simp
+        rw [hn]
+        simp only [smul_assoc, map_smul, chainPackaging_symm_translate, RingHom.id_apply]
+        rw [hA g _, chainPackaging_translate]
+
+@[simp] theorem chainRingMap_apply (A : BasedChains Γ S →ₗ[ℤ] BasedChains Γ T)
+    (hA : ChainsEquivariant A) (c : S →₀ IntegralGroupRing Γ) :
+    chainRingMap A hA c = chainPackaging (A (chainPackaging.symm c)) := rfl
+
+theorem chainPackaging_mem_cycles (v : S → Γ) (c : BasedChains Γ S) :
+    c ∈ cycles v ↔ chainPackaging c ∈ LinearMap.ker (cayleyDifferential v) := by
+  change edgeBoundary v c = 0 ↔ cayleyDifferential v (chainPackaging c) = 0
+  rw [chainPackaging_boundary]
+  constructor
+  · intro h
+    rw [h]
+    rfl
+  · intro h
+    have hh := congrArg MonoidAlgebra.coeff h
+    exact hh
+
+theorem projective_cayley_cycles_of_boundary_basis (v : S → Γ)
+    (A : BasedChains Γ T →ₗ[ℤ] BasedChains Γ S) (hA : ChainsEquivariant A)
+    (hAi : Function.Injective A) (hAr : LinearMap.range A = cycles v) :
+    Module.Projective (IntegralGroupRing Γ) (LinearMap.ker (cayleyDifferential v)) := by
+  have hm : ∀ c, chainRingMap A hA c ∈ LinearMap.ker (cayleyDifferential v) := by
+    intro c
+    apply (chainPackaging_mem_cycles v _).mp
+    rw [← hAr]
+    exact ⟨chainPackaging.symm c,rfl⟩
+  let d := (chainRingMap A hA).codRestrict _ hm
+  have hi : Function.Injective d := by
+    intro c c' h
+    apply (chainPackaging (Γ := Γ) (S := T)).symm.injective
+    apply hAi
+    apply chainPackaging.injective
+    exact congrArg Subtype.val h
+  have hs : Function.Surjective d := by
+    intro c
+    have hc : chainPackaging.symm (c : S →₀ IntegralGroupRing Γ) ∈ cycles v := by
+      apply (chainPackaging_mem_cycles v _).mpr
+      rw [LinearEquiv.apply_symm_apply]
+      exact c.property
+    rw [← hAr] at hc
+    obtain ⟨b,hb⟩ := hc
+    refine ⟨chainPackaging b,Subtype.ext ?_⟩
+    change chainPackaging (A (chainPackaging.symm (chainPackaging b))) = c.val
+    rw [LinearEquiv.symm_apply_apply,hb,LinearEquiv.apply_symm_apply]
+  exact Module.Projective.of_equiv' (LinearEquiv.ofBijective d ⟨hi,hs⟩)
+
+theorem projectiveDimension_le_two_of_boundary_basis (v : S → Γ)
+    (hv : Function.Surjective (FreeGroup.lift v))
+    (A : BasedChains Γ T →ₗ[ℤ] BasedChains Γ S) (hA : ChainsEquivariant A)
+    (hAi : Function.Injective A) (hAr : LinearMap.range A = cycles v) :
+    CategoryTheory.projectiveDimension
+      (ModuleCat.of (IntegralGroupRing Γ) (IntegralTrivialModule Γ)) ≤ 2 := by
+  have := projective_cayley_cycles_of_boundary_basis v A hA hAi hAr
+  exact cayley_projectiveDimension_le_two v hv
+
+end ChainPackaging
+
+
+end EilenbergGanea
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/LevelBasis.lean

@@ -1,968 +1,968 @@
-import OAI.AlgebraicTopology.Cubical.HeightChains
-import OAI.Topology.EilenbergGanea.IntegralModules
-import OAI.Topology.EilenbergGanea.LevelTransport
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set Filter Topology MeasureTheory
-open scoped Quaternion ContDiff
-
-namespace EilenbergGanea.LevelBasis
-open CubicalChains CubicalChains.LevelExact CubicalChains.HeightFiltration
-variable {G A : Type} [Group G] [LinearOrder A]
-variable (L : SimpleGraph A) (φ : G →* Multiplicative ℤ) (v : A → G) (root : A)
-def ht (g : G) : ℤ := (φ g).toAdd
-@[simp] theorem ht_mul (g q : G) : ht φ (g*q) = ht φ g + ht φ q := by simp [ht]
-@[simp] theorem ht_one : ht φ 1 = 0 := by simp [ht]
-@[simp] theorem ht_inv (g : G) : ht φ g⁻¹ = -ht φ g := by simp [ht]
-@[simp] theorem ht_zpow (g : G) (n : ℤ) : ht φ (g^n) = n * ht φ g := by simp [ht]
-@[simp] theorem ht_ker (g : φ.ker) : ht φ g = 0 := by simp [ht,MonoidHom.mem_ker.mp g.property]
-variable (hv : ∀ a, φ (v a) = Multiplicative.ofAdd (1 : ℤ))
-include hv in
-omit [LinearOrder A] in
-@[simp] theorem ht_generator (a : A) : ht φ (v a) = 1 := by simp [ht,hv]
-include hv in
-omit [LinearOrder A] in
-theorem ht_step (g : G) (a : A) : ht φ (g * v a) = ht φ g + 1 := by simp [ht_generator φ v hv]
-
-def Shape (k : ℕ) := {p : List A × ℤ // SimplicialChains.Valid L p.1 ∧ p.1.length = k ∧
-  p.2 < 0 ∧ 0 < p.2 + (k : ℤ)}
-def basisIndex (k : ℕ) (p : φ.ker × Shape L k) : G × List A :=
-  (p.1.val * (v (p.2.val.1.headD root)) ^ p.2.val.2,p.2.val.1)
-
-include hv in
-theorem basisIndex_height (k : ℕ) (p : φ.ker × Shape L k) :
-    ht φ (basisIndex L φ v root k p).1 = p.2.val.2 := by
-  simp only [basisIndex,ht_mul,ht_ker,ht_zpow,ht_generator φ v hv,mul_one,zero_add]
-
-include hv in
-theorem basisIndex_injective (k : ℕ) : Function.Injective (basisIndex L φ v root k) := by
-  intro p q he
-  have hw : p.2.val.1 = q.2.val.1 := congrArg Prod.snd he
-  have hh := congrArg (fun x => ht φ x.1) he
-  rw [basisIndex_height L φ v root hv,basisIndex_height L φ v root hv] at hh
-  have hs : p.2 = q.2 := Subtype.ext (Prod.ext hw hh)
-  have hg := congrArg Prod.fst he
-  simp only [basisIndex,hs] at hg
-  exact Prod.ext (Subtype.ext (mul_right_cancel hg)) hs
-
-include hv in
-theorem basisIndex_mem (k : ℕ) (p : φ.ker × Shape L k) :
-    SimplicialChains.Valid L (basisIndex L φ v root k p).2 ∧
-      (basisIndex L φ v root k p).2.length = k ∧
-      ht φ (basisIndex L φ v root k p).1 < 0 ∧
-      0 < ht φ (basisIndex L φ v root k p).1 + (k : ℤ) := by
-  rw [basisIndex_height L φ v root hv]
-  exact p.2.property
-
-include hv in
-theorem exists_basisIndex {k : ℕ} (p : G × List A)
-    (hp : SimplicialChains.Valid L p.2 ∧ p.2.length = k ∧ ht φ p.1 < 0 ∧ 0 < ht φ p.1 + (k : ℤ)) :
-    ∃ q : φ.ker × Shape L k, basisIndex L φ v root k q = p := by
-  let s : Shape L k := ⟨(p.2,ht φ p.1),hp⟩
-  let g : φ.ker := ⟨p.1 * (v (p.2.headD root)) ^ (-ht φ p.1),by
-    apply (Multiplicative.toAdd.injective)
-    change ht φ (p.1 * (v (p.2.headD root)) ^ (-ht φ p.1)) = 0
-    simp only [ht_mul,ht_zpow,ht_generator φ v hv,mul_one,add_neg_cancel]⟩
-  refine ⟨(g,s),?_⟩
-  apply Prod.ext
-  · change (p.1 * (v (p.2.headD root)) ^ (-ht φ p.1)) * (v (p.2.headD root)) ^ (ht φ p.1) = p.1
-    rw [mul_assoc,← zpow_add,neg_add_cancel,zpow_zero,mul_one]
-  · rfl
-
-def includeBasis (k : ℕ) : (φ.ker × Shape L k →₀ ℤ) →ₗ[ℤ] CubicalChains.Chains G A :=
-  Finsupp.lmapDomain ℤ ℤ (basisIndex L φ v root k)
-@[simp] theorem includeBasis_single (k : ℕ) (p : φ.ker × Shape L k) (n : ℤ) :
-    includeBasis L φ v root k (Finsupp.single p n) = Finsupp.single (basisIndex L φ v root k p) n := by
-  simp [includeBasis]
-include hv in
-theorem includeBasis_injective (k : ℕ) : Function.Injective (includeBasis L φ v root k) :=
-  Finsupp.mapDomain_injective (basisIndex_injective L φ v root hv k)
-
-include hv in
-theorem includeBasis_range (k : ℕ) : LinearMap.range (includeBasis L φ v root k) = crossing L (ht φ) k := by
-  rw [includeBasis,Finsupp.range_lmapDomain,crossing,Finsupp.supported_eq_span_single]
-  congr 1
-  ext c
-  constructor
-  · rintro ⟨p,rfl⟩
-    exact ⟨basisIndex L φ v root k p,basisIndex_mem L φ v root hv k p,rfl⟩
-  · rintro ⟨p,hp,rfl⟩
-    obtain ⟨q,rfl⟩ := exists_basisIndex L φ v root hv p hp
-    exact ⟨q,rfl⟩
-
-variable (comm : ∀ a b, L.Adj a b → Commute (v a) (v b))
-include hv in
-def projectBasis (k : ℕ) : CubicalChains.Chains G A →ₗ[ℤ] (φ.ker × Shape L k →₀ ℤ) :=
-  Finsupp.lcomapDomain (basisIndex L φ v root k) (basisIndex_injective L φ v root hv k)
-
-@[simp] theorem project_include (k : ℕ) (c : φ.ker × Shape L k →₀ ℤ) :
-    projectBasis L φ v root hv k (includeBasis L φ v root k c) = c :=
-  Finsupp.leftInverse_lcomapDomain_mapDomain _ _ c
-
-include hv in
-theorem include_project {k : ℕ} {c : CubicalChains.Chains G A} (hc : c ∈ crossing L (ht φ) k) :
-    includeBasis L φ v root k (projectBasis L φ v root hv k c) = c := by
-  rw [← includeBasis_range L φ v root hv k] at hc
-  obtain ⟨d,rfl⟩ := hc
-  rw [project_include]
-
-include hv in
-theorem includeBasis_mem (k : ℕ) (c : φ.ker × Shape L k →₀ ℤ) :
-    includeBasis L φ v root k c ∈ crossing L (ht φ) k := by
-  rw [← includeBasis_range L φ v root hv k]
-  exact ⟨c,rfl⟩
-
-/-- The level edge parameter has precisely two directions and initial height -1. -/
-theorem shape_two_form (s : Shape L 2) : ∃ a b, s.val = ([a,b],-1) := by
-  obtain ⟨a,b,hab⟩ := List.length_eq_two.mp s.property.2.1
-  refine ⟨a,b,Prod.ext hab ?_⟩
-  have hn := s.property.2.2.1
-  have hm := s.property.2.2.2
-  norm_num at hm
-  omega
-
-def generator (s : Shape L 2) : φ.ker :=
-  ⟨(v (s.val.1.headD root))⁻¹ * v (s.val.1.tail.headD root),by simp [hv]⟩
-
-def includeVertices : (φ.ker →₀ ℤ) →ₗ[ℤ] CubicalChains.Chains G A :=
-  Finsupp.lmapDomain ℤ ℤ (fun g => (g.val,[]))
-omit [LinearOrder A] in
-@[simp] theorem includeVertices_single (g : φ.ker) (n : ℤ) :
-    includeVertices (A := A) φ (Finsupp.single g n) = Finsupp.single (g.val,[]) n := by simp [includeVertices]
-omit [LinearOrder A] in
-theorem includeVertices_injective : Function.Injective (includeVertices (A := A) φ) := by
-  apply Finsupp.mapDomain_injective
-  intro p q he
-  exact Subtype.ext (congrArg Prod.fst he)
-
-include hv in
-omit [LinearOrder A] in
-theorem boundary_square (g : G) (a b : A) (hab : Commute (v a) (v b))
-    (hg : ht φ g = -1) :
-    levelBoundary v (ht φ) (Finsupp.single (g,[a,b]) 1) =
-      Finsupp.single (g*v b,[]) 1 - Finsupp.single (g*v a,[]) 1 := by
-  have hga : ht φ (g * v a) = 0 := by rw [ht_step φ v hv,hg]; omega
-  have hgb : ht φ (g * v b) = 0 := by rw [ht_step φ v hv,hg]; omega
-  simp only [levelBoundary_apply,boundary_single,one_smul,boundaryTerm_cons,boundaryTerm_nil,
-    map_zero,sub_zero,map_sub,front_single,plus,select_single,hga,hgb,hg]
-  norm_num
-  rw [mul_assoc,hab.eq,← mul_assoc]
-  abel
-
-
-include hv comm in
-theorem levelBoundary_include_single (p : φ.ker × Shape L 2) :
-    levelBoundary v (ht φ) (includeBasis L φ v root 2 (Finsupp.single p 1)) =
-      includeVertices (A := A) φ (edgeBoundary (generator L φ v root hv) (Finsupp.single p 1)) := by
-  rcases p with ⟨g,⟨⟨w,n⟩,hs⟩⟩
-  obtain ⟨a,b,hw⟩ := List.length_eq_two.mp hs.2.1
-  change w = [a,b] at hw
-  subst w
-  have hn : n = -1 := by have h₁ := hs.2.2.1; have h₂ := hs.2.2.2; norm_num at h₂; omega
-  subst n
-  have hab : L.Adj a b := (List.pairwise_cons.mp hs.1.2).1 b (by simp)
-  rw [includeBasis_single]
-  change levelBoundary v (ht φ) (Finsupp.single (g.val * (v a)^(-1 : ℤ),[a,b]) 1) = _
-  rw [boundary_square φ v hv _ a b (comm a b hab) (by simp [ht_generator φ v hv])]
-  simp only [edgeBoundary,Finsupp.linearCombination_single]
-  simp only [zpow_neg_one,one_smul,map_sub,includeVertices_single,
-    Subgroup.coe_mul,generator,List.headD_cons,List.tail_cons,mul_assoc,inv_mul_cancel,mul_one]
-
-include hv comm in
-theorem levelBoundary_include (c : φ.ker × Shape L 2 →₀ ℤ) :
-    levelBoundary v (ht φ) (includeBasis L φ v root 2 c) =
-      includeVertices (A := A) φ (edgeBoundary (generator L φ v root hv) c) := by
-  induction c using Finsupp.induction_linear with
-  | zero => simp
-  | add c d hc hd => simp only [map_add,hc,hd]
-  | single p n =>
-    have he : Finsupp.single p n = n • Finsupp.single p (1 : ℤ) := by simp
-    rw [he,map_smul,map_smul,map_smul,map_smul,levelBoundary_include_single L φ v root hv comm]
-
-/-- The literal based integral boundary of the crossing three-cubes. -/
-def triangleBoundary : (φ.ker × Shape L 3 →₀ ℤ) →ₗ[ℤ] (φ.ker × Shape L 2 →₀ ℤ) :=
-  (projectBasis L φ v root hv 2).comp ((crossPart (ht φ)).comp ((boundary v).comp (includeBasis L φ v root 3)))
-
-include hv in
-theorem triangleBoundary_inclusion (c : φ.ker × Shape L 3 →₀ ℤ) :
-    includeBasis L φ v root 2 (triangleBoundary L φ v root hv c) =
-      crossPart (ht φ) (boundary v (includeBasis L φ v root 3 c)) := by
-  apply include_project L φ v root hv
-  exact crossPart_mem L (ht φ) (boundary_total L v (ht φ) (ht_step φ v hv)
-    (crossing_total L (ht φ) (includeBasis_mem L φ v root hv 3 c)))
-
-include hv comm in
-theorem triangleBoundary_cycle (c : φ.ker × Shape L 3 →₀ ℤ) :
-    triangleBoundary L φ v root hv c ∈ cycles (generator L φ v root hv) := by
-  apply includeVertices_injective φ
-  rw [map_zero,← levelBoundary_include L φ v root hv comm,triangleBoundary_inclusion L φ v root hv]
-  exact levelBoundary_cross_boundary L v (ht φ) (ht_step φ v hv) comm (includeBasis_mem L φ v root hv 3 c)
-
-variable (K : SimplicialChains.Contraction L)
-variable (exactTotal : ∀ {k : ℕ}, 0 < k → ∀ c, c ∈ total (G := G) L k → boundary v c = 0 →
-      ∃ b ∈ total (G := G) L (k+1), boundary v b = c)
-variable (dim : ∀ w, SimplicialChains.Valid L w → w.length ≤ 3)
-include hv comm K exactTotal in
-theorem triangleBoundary_range : LinearMap.range (triangleBoundary L φ v root hv) =
-    cycles (generator L φ v root hv) := by
-  apply le_antisymm
-  · rintro _ ⟨c,rfl⟩
-    exact triangleBoundary_cycle L φ v root hv comm c
-  · intro c hc
-    have hz : levelBoundary v (ht φ) (includeBasis L φ v root 2 c) = 0 := by
-      rw [levelBoundary_include L φ v root hv comm,LinearMap.mem_ker.mp hc,map_zero]
-    obtain ⟨t,ht,hdt⟩ := level_cycle_bounds L v (ht φ) K (ht_step φ v hv) comm exactTotal
-      _ (includeBasis_mem L φ v root hv 2 c) hz
-    rw [← includeBasis_range L φ v root hv 3] at ht
-    obtain ⟨b,rfl⟩ := ht
-    refine ⟨b,includeBasis_injective L φ v root hv 2 ?_⟩
-    rwa [triangleBoundary_inclusion L φ v root hv]
-
-include hv comm K exactTotal dim in
-theorem triangleBoundary_injective : Function.Injective (triangleBoundary L φ v root hv) := by
-  suffices hz : ∀ c, triangleBoundary L φ v root hv c = 0 → c = 0 by
-    intro c d he
-    exact sub_eq_zero.mp (hz (c-d) (by simp only [map_sub,he,sub_self]))
-  intro c hc
-  apply includeBasis_injective L φ v root hv 3
-  rw [map_zero]
-  apply level_boundary_injective L v (ht φ) K (ht_step φ v hv) comm exactTotal dim
-    _ (includeBasis_mem L φ v root hv 3 c)
-  rw [← triangleBoundary_inclusion L φ v root hv,hc,map_zero]
-
-
-/-- Left translation on the actual ambient cube chains. -/
-def leftTranslate (g : G) : CubicalChains.Chains G A →ₗ[ℤ] CubicalChains.Chains G A :=
-  Finsupp.lmapDomain ℤ ℤ (fun p => (g*p.1,p.2))
-omit [LinearOrder A] in
-@[simp] theorem leftTranslate_single (g q : G) (w : List A) (n : ℤ) :
-    leftTranslate g (Finsupp.single (q,w) n) = Finsupp.single (g*q,w) n := by simp [leftTranslate]
-omit [LinearOrder A] in
-theorem leftTranslate_front (g : G) (a : A) (c : CubicalChains.Chains G A) :
-    leftTranslate g (CubicalChains.front a c) = CubicalChains.front a (leftTranslate g c) := by
-  induction c using Finsupp.induction_linear with
-  | zero => simp
-  | add c d hc hd => simp only [map_add,hc,hd]
-  | single p n => rcases p with ⟨q,w⟩; simp
-omit [LinearOrder A] in
-theorem leftTranslate_boundaryTerm (g q : G) (w : List A) :
-    leftTranslate g (boundaryTerm v q w) = boundaryTerm v (g*q) w := by
-  induction w with
-  | nil => simp
-  | cons a w ih => simp only [boundaryTerm_cons,map_sub,leftTranslate_single,leftTranslate_front,ih,mul_assoc]
-omit [LinearOrder A] in
-theorem leftTranslate_boundary (g : G) (c : CubicalChains.Chains G A) :
-    leftTranslate g (boundary v c) = boundary v (leftTranslate g c) := by
-  induction c using Finsupp.induction_linear with
-  | zero => simp
-  | add c d hc hd => simp only [map_add,hc,hd]
-  | single p n => rcases p with ⟨q,w⟩; simp [leftTranslate_boundaryTerm]
-omit [LinearOrder A] in
-theorem leftTranslate_crossPart (g : φ.ker) (c : CubicalChains.Chains G A) :
-    leftTranslate g.val (crossPart (ht φ) c) = crossPart (ht φ) (leftTranslate g.val c) := by
-  induction c using Finsupp.induction_linear with
-  | zero => simp
-  | add c d hc hd => simp only [map_add,hc,hd]
-  | single p n =>
-      rcases p with ⟨q,w⟩
-      simp only [crossPart,select_single,leftTranslate_single,ht_mul,ht_ker,zero_add]
-      split <;> simp_all
-
-theorem includeBasis_translate (k : ℕ) (g : φ.ker) (c : φ.ker × Shape L k →₀ ℤ) :
-    includeBasis L φ v root k (translateChains g c) = leftTranslate g.val (includeBasis L φ v root k c) := by
-  induction c using Finsupp.induction_linear with
-  | zero => simp
-  | add c d hc hd => simp only [map_add,hc,hd]
-  | single p n =>
-      rcases p with ⟨q,s⟩
-      simp only [translateChains_single,includeBasis_single,leftTranslate_single,basisIndex,
-        Subgroup.coe_mul,mul_assoc]
-
-include hv in
-theorem triangleBoundary_equivariant : ChainsEquivariant (triangleBoundary L φ v root hv) := by
-  intro g c
-  apply includeBasis_injective L φ v root hv 2
-  rw [triangleBoundary_inclusion L φ v root hv,includeBasis_translate,
-    ← leftTranslate_boundary,← leftTranslate_crossPart,includeBasis_translate,
-    triangleBoundary_inclusion L φ v root hv]
-
-end EilenbergGanea.LevelBasis
-
-namespace EilenbergGanea.LevelBasis
-variable {V : Type} [LinearOrder V] (L : SimpleGraph V)
-
-def pairShape (a b : V) (hab : a < b) (hadj : L.Adj a b) : Shape L 2 :=
-  ⟨([a,b],-1),by simp [SimplicialChains.Valid,hab,hadj]⟩
-
-@[simp] theorem generator_pair (root a b : V) (hab : a < b) (hadj : L.Adj a b) :
-    (generator L (height L) (artinGenerator L) root (height_generator L)
-      (pairShape L a b hab hadj)).val = (artinGenerator L a)⁻¹ * artinGenerator L b := rfl
-
-/-- The actual oriented crossing-square edges generate precisely the height kernel. -/
-theorem generator_surjective (hconn : L.Preconnected) (root : V) :
-    Function.Surjective (FreeGroup.lift (generator L (height L) (artinGenerator L) root (height_generator L))) := by
-  let f := FreeGroup.lift (generator L (height L) (artinGenerator L) root (height_generator L))
-  let F := (height L).ker.subtype.comp f
-  have hgen : edgeGenerated L ≤ F.range := by
-    apply (Subgroup.closure_le _).mpr
-    rintro x ⟨a,b,hadj,rfl⟩
-    have hab : a ≠ b := L.ne_of_adj hadj
-    rcases lt_or_gt_of_ne hab with hab | hab
-    · refine ⟨(FreeGroup.of (pairShape L a b hab hadj))⁻¹,?_⟩
-      change (f ((FreeGroup.of (pairShape L a b hab hadj))⁻¹)).val = _
-      simp only [f,map_inv,FreeGroup.lift_apply_of,Subgroup.coe_inv,generator_pair,mul_inv_rev,inv_inv]
-      exact (adjacent_generators_commute L hadj).inv_right.eq.symm
-    · refine ⟨FreeGroup.of (pairShape L b a hab hadj.symm),?_⟩
-      change (f (FreeGroup.of (pairShape L b a hab hadj.symm))).val = _
-      simp only [f,FreeGroup.lift_apply_of,generator_pair]
-      exact (adjacent_generators_commute L hadj).inv_right.eq.symm
-  intro g
-  have he : g.val ∈ edgeGenerated L := by rw [edgeGenerated_eq_height_kernel L hconn root]; exact g.property
-  obtain ⟨w,hw⟩ := hgen he
-  exact ⟨w,Subtype.ext hw⟩
-
-end EilenbergGanea.LevelBasis
-
-namespace EilenbergGanea.LevelBasis
-open CubicalChains CubicalChains.HeightFiltration
-variable {V : Type} [LinearOrder V] [Fintype V] [Nonempty V] (L : SimpleGraph V)
-
-/-- The exactness input to the level slicing is the actual RAAG cube-chain
-contraction, not an assumption about a classifying space. -/
-theorem artin_total_cycle_bounds {k : ℕ} (hk : 0 < k)
-    (c : Chains (ArtinGroup L) V) (hc : c ∈ total (G := ArtinGroup L) L k)
-    (hz : boundary (artinGenerator L) c = 0) :
-    ∃ b ∈ total (G := ArtinGroup L) L (k+1), boundary (artinGenerator L) b = c := by
-  have he : ∀ n, total (G := ArtinGroup L) L n = TraceWords.CubeDescents.positiveChains L n := by
-    intro n
-    simp only [total,TraceWords.CubeDescents.positiveChains,SimplicialChains.Valid,and_assoc]
-  rw [he] at hc ⊢
-  exact TraceWords.CubeDescents.positive_cycle_bounds L hk c hc hz
-
-/-- Integral cd at most two for the genuine height kernel, from a genuine
-finite augmented link contraction and the dimension of its flag complex. -/
-theorem artin_height_kernel_cd_le_two (root : V) (hconn : L.Preconnected)
-    (K : SimplicialChains.Contraction L)
-    (dim : ∀ w, SimplicialChains.Valid L w → w.length ≤ 3) :
-    cohomologicalDimension (height L).ker ≤ 2 := by
-  exact projectiveDimension_le_two_of_boundary_basis
-    (generator L (height L) (artinGenerator L) root (height_generator L))
-    (generator_surjective L hconn root)
-    (triangleBoundary L (height L) (artinGenerator L) root (height_generator L))
-    (triangleBoundary_equivariant L (height L) (artinGenerator L) root (height_generator L))
-    (triangleBoundary_injective L (height L) (artinGenerator L) root (height_generator L)
-      (fun _ _ h => adjacent_generators_commute L h) K (artin_total_cycle_bounds L) dim)
-    (triangleBoundary_range L (height L) (artinGenerator L) root (height_generator L)
-      (fun _ _ h => adjacent_generators_commute L h) K (artin_total_cycle_bounds L))
-end EilenbergGanea.LevelBasis
-namespace EilenbergGanea.LevelBasis
-open CubicalChains CubicalChains.LevelExact CubicalChains.HeightFiltration
-variable {G A : Type} [Group G] [LinearOrder A]
-variable (L : SimpleGraph A) (φ : G →* Multiplicative ℤ) (v : A → G) (root : A)
-variable (hv : ∀ a, φ (v a) = Multiplicative.ofAdd (1 : ℤ))
-variable (comm : ∀ a b, L.Adj a b → Commute (v a) (v b))
-
-instance shape_finite [Fintype A] (k : ℕ) : Finite (Shape L k) := by
-  let f : Shape L k → List.Vector A k × Fin k := fun s =>
-    (⟨s.val.1,s.property.2.1⟩,⟨(-s.val.2).toNat,by
-      have hn := s.property.2.2.1
-      have hm := s.property.2.2.2
-      omega⟩)
-  apply Finite.of_injective f
-  intro s t h
-  apply Subtype.ext
-  apply Prod.ext
-  · exact congrArg (fun p : List.Vector A k × Fin k => p.1.val) h
-  · have he := congrArg (fun p : List.Vector A k × Fin k => p.2.val) h
-    have hs := s.property.2.2.1
-    have ht := t.property.2.2.1
-    dsimp [f] at he
-    omega
-
-noncomputable instance shape_fintype [Fintype A] (k : ℕ) : Fintype (Shape L k) := Fintype.ofFinite _
-
-def edgeShape (a b : A) (hab : SimplicialChains.Valid L [a,b]) : Shape L 2 :=
-  ⟨([a,b],-1),hab,rfl,by norm_num,by norm_num⟩
-
-def triangleShape (a b c : A) (habc : SimplicialChains.Valid L [a,b,c])
-    (n : ℤ) (hn : n = -1 ∨ n = -2) : Shape L 3 :=
-  ⟨([a,b,c],n),habc,rfl,by omega,by norm_num; omega⟩
-
-@[simp] theorem generator_edgeShape (a b : A) (hab : SimplicialChains.Valid L [a,b]) :
-    (generator L φ v root hv (edgeShape L a b hab) : G) = (v a)⁻¹ * v b := rfl
-
-theorem valid_triangle_edges {a b c : A} (h : SimplicialChains.Valid L [a,b,c]) :
-    SimplicialChains.Valid L [a,b] ∧ SimplicialChains.Valid L [a,c] ∧
-      SimplicialChains.Valid L [b,c] := by
-  simp [SimplicialChains.Valid] at h ⊢
-  tauto
-
-include comm in
-theorem edge_values_compose (a b c : A) (h : SimplicialChains.Valid L [a,b,c]) :
-    ((v a)⁻¹ * v b) * ((v b)⁻¹ * v c) = (v a)⁻¹ * v c ∧
-    ((v b)⁻¹ * v c) * ((v a)⁻¹ * v b) = (v a)⁻¹ * v c := by
-  have he := valid_triangle_edges L h
-  have ha : L.Adj a b := by simpa [SimplicialChains.Valid] using he.1.2
-  have hb : L.Adj a c := by simpa [SimplicialChains.Valid] using he.2.1.2
-  have hc : L.Adj b c := by simpa [SimplicialChains.Valid] using he.2.2.2
-  constructor
-  · group
-  · calc
-      ((v b)⁻¹ * v c) * ((v a)⁻¹ * v b) = (v a)⁻¹ * ((v b)⁻¹ * v c) * v b := by
-        rw [← mul_assoc,
-          ((comm a b ha).inv_left.inv_right.mul_right (comm a c hb).inv_left).symm.eq]
-      _ = (v a)⁻¹ * v c := by
-        rw [mul_assoc,mul_assoc,← (comm b c hc).eq,inv_mul_cancel_left]
-
-def lowerTriangleWord (ab ac bc : Shape L 2) : FreeGroup (Shape L 2) :=
-  FreeGroup.of ac * (FreeGroup.of bc)⁻¹ * (FreeGroup.of ab)⁻¹
-
-def upperTriangleWord (ab ac bc : Shape L 2) : FreeGroup (Shape L 2) :=
-  FreeGroup.of ab * (FreeGroup.of bc * FreeGroup.of ab * (FreeGroup.of ac)⁻¹) *
-    (FreeGroup.of ab)⁻¹
-
-include comm in
-theorem triangle_words_trivial (a b c : A) (h : SimplicialChains.Valid L [a,b,c]) :
-    let e := valid_triangle_edges L h
-    let ab := edgeShape L a b e.1
-    let ac := edgeShape L a c e.2.1
-    let bc := edgeShape L b c e.2.2
-    wordValue (generator L φ v root hv) (lowerTriangleWord L ab ac bc) = 1 ∧
-      wordValue (generator L φ v root hv) (upperTriangleWord L ab ac bc) = 1 := by
-  dsimp only
-  have he := edge_values_compose L v comm a b c h
-  constructor <;> apply Subtype.ext
-  · simp only [lowerTriangleWord,map_mul,map_inv,wordValue,FreeGroup.lift_apply_of,
-      Subgroup.coe_mul,Subgroup.coe_inv,generator_edgeShape,OneMemClass.coe_one]
-    group
-  · simp only [upperTriangleWord,map_mul,map_inv,wordValue,FreeGroup.lift_apply_of,
-      Subgroup.coe_mul,Subgroup.coe_inv,generator_edgeShape,OneMemClass.coe_one]
-    rw [he.2,mul_inv_cancel,mul_one,mul_inv_cancel]
-
-include hv in
-theorem triangle_boundary_low (g : φ.ker) (a b c : A)
-    (h : SimplicialChains.Valid L [a,b,c]) :
-    let e := valid_triangle_edges L h
-    let ab := edgeShape L a b e.1
-    let ac := edgeShape L a c e.2.1
-    let bc := edgeShape L b c e.2.2
-    triangleBoundary L φ v root hv
-        (Finsupp.single (g,triangleShape L a b c h (-1) (Or.inl rfl)) 1) =
-      Finsupp.single (g,ac) 1 - Finsupp.single (g,ab) 1 -
-        Finsupp.single (g * generator L φ v root hv ab,bc) 1 := by
-  dsimp only
-  apply includeBasis_injective L φ v root hv 2
-  rw [triangleBoundary_inclusion L φ v root hv]
-  simp only [map_sub,includeBasis_single,basisIndex,triangleShape,edgeShape,
-    List.headD_cons,zpow_neg_one,Subgroup.coe_mul,generator,List.tail_cons]
-  simp only [boundary_single,one_smul,boundaryTerm_cons,boundaryTerm_nil,map_zero,
-    sub_zero,map_sub,front_single,crossPart,select,Finsupp.linearCombination_single,
-    ht_mul,ht_ker,ht_inv,ht_generator φ v hv,List.length_cons,List.length_nil]
-  norm_num
-  simp only [mul_assoc,mul_inv_cancel,mul_one]
-  abel
-
-include hv comm in
-theorem triangle_boundary_high (g : φ.ker) (a b c : A)
-    (h : SimplicialChains.Valid L [a,b,c]) :
-    let e := valid_triangle_edges L h
-    let ab := edgeShape L a b e.1
-    let ac := edgeShape L a c e.2.1
-    let bc := edgeShape L b c e.2.2
-    triangleBoundary L φ v root hv
-        (Finsupp.single (g,triangleShape L a b c h (-2) (Or.inr rfl)) 1) =
-      Finsupp.single (g * generator L φ v root hv ab,bc) 1 -
-        Finsupp.single (g * generator L φ v root hv ab,ac) 1 +
-        Finsupp.single (g * generator L φ v root hv ac,ab) 1 := by
-  dsimp only
-  have eh := valid_triangle_edges L h
-  have hab : L.Adj a b := by simpa [SimplicialChains.Valid] using eh.1.2
-  have hac : L.Adj a c := by simpa [SimplicialChains.Valid] using eh.2.1.2
-  apply includeBasis_injective L φ v root hv 2
-  rw [triangleBoundary_inclusion L φ v root hv]
-  simp only [map_sub,map_add,includeBasis_single,basisIndex,triangleShape,edgeShape,
-    List.headD_cons,Subgroup.coe_mul,generator,List.tail_cons]
-  simp only [boundary_single,one_smul,boundaryTerm_cons,boundaryTerm_nil,map_zero,
-    sub_zero,map_sub,front_single,crossPart,select,map_sub,Finsupp.linearCombination_single,
-    ht_mul,ht_ker,ht_zpow,ht_generator φ v hv,List.length_cons,List.length_nil]
-  norm_num
-  have hb : (v a)^(-2 : ℤ) * v b = (v a)⁻¹ * v b * (v a)⁻¹ := by
-    calc
-      (v a)^(-2 : ℤ) * v b = (v a)⁻¹ * ((v a)⁻¹ * v b) := by group
-      _ = (v a)⁻¹ * (v b * (v a)⁻¹) := congrArg ((v a)⁻¹ * ·) (comm a b hab).inv_left.eq
-      _ = _ := (mul_assoc _ _ _).symm
-  have hc : (v a)^(-2 : ℤ) * v c = (v a)⁻¹ * v c * (v a)⁻¹ := by
-    calc
-      (v a)^(-2 : ℤ) * v c = (v a)⁻¹ * ((v a)⁻¹ * v c) := by group
-      _ = (v a)⁻¹ * (v c * (v a)⁻¹) := congrArg ((v a)⁻¹ * ·) (comm a c hac).inv_left.eq
-      _ = _ := (mul_assoc _ _ _).symm
-  have ha : (v a)^(-2 : ℤ) * v a = (v a)⁻¹ := by group
-  simp only [zpow_neg,zpow_ofNat] at ha hb hc
-  simp only [mul_assoc] at hb hc
-  simp only [mul_assoc,ha,hb,hc,mul_inv_cancel,mul_one]
-  abel
-
-section WordChains
-variable (w : Shape L 2 → G) (ab ac bc : Shape L 2)
-theorem lowerTriangleWord_chain (habc : w ab * w bc = w ac) (g : G) :
-    pathChain w (lowerTriangleWord L ab ac bc) g =
-      Finsupp.single (g,ac) 1 - Finsupp.single (g,ab) 1 -
-        Finsupp.single (g * w ab,bc) 1 := by
-  simp only [lowerTriangleWord,pathChain_mul,pathChain_inv,pathChain_of,
-    map_mul,map_inv,wordValue,FreeGroup.lift_apply_of]
-  rw [← habc]
-  simp only [mul_assoc,mul_inv_cancel,mul_one]
-  abel
-
-theorem upperTriangleWord_chain (habc : w ab * w bc = w ac)
-    (hbac : w bc * w ab = w ac) (g : G) :
-    pathChain w (upperTriangleWord L ab ac bc) g =
-      Finsupp.single (g * w ab,bc) 1 - Finsupp.single (g * w ab,ac) 1 +
-        Finsupp.single (g * w ac,ab) 1 := by
-  have hr : wordValue w (FreeGroup.of bc * FreeGroup.of ab * (FreeGroup.of ac)⁻¹) = 1 := by
-    simp only [wordValue,map_mul,map_inv,FreeGroup.lift_apply_of,hbac,mul_inv_cancel]
-  rw [upperTriangleWord,pathChain_conjugate w _ _ hr]
-  simp only [pathChain_mul,pathChain_inv,pathChain_of,map_mul,wordValue,
-    FreeGroup.lift_apply_of]
-  rw [hbac]
-  simp only [mul_assoc,habc,mul_inv_cancel,mul_one]
-  abel
-end WordChains
-
-include comm in
-theorem triangle_words_chain (g : φ.ker) (a b c : A) (h : SimplicialChains.Valid L [a,b,c]) :
-    let e := valid_triangle_edges L h
-    let ab := edgeShape L a b e.1
-    let ac := edgeShape L a c e.2.1
-    let bc := edgeShape L b c e.2.2
-    pathChain (generator L φ v root hv) (lowerTriangleWord L ab ac bc) g =
-      triangleBoundary L φ v root hv
-        (Finsupp.single (g,triangleShape L a b c h (-1) (Or.inl rfl)) 1) ∧
-    pathChain (generator L φ v root hv) (upperTriangleWord L ab ac bc) g =
-      triangleBoundary L φ v root hv
-        (Finsupp.single (g,triangleShape L a b c h (-2) (Or.inr rfl)) 1) := by
-  dsimp only
-  have he := edge_values_compose L v comm a b c h
-  have h₁ : generator L φ v root hv (edgeShape L a b (valid_triangle_edges L h).1) *
-      generator L φ v root hv (edgeShape L b c (valid_triangle_edges L h).2.2) =
-      generator L φ v root hv (edgeShape L a c (valid_triangle_edges L h).2.1) :=
-    Subtype.ext he.1
-  have h₂ : generator L φ v root hv (edgeShape L b c (valid_triangle_edges L h).2.2) *
-      generator L φ v root hv (edgeShape L a b (valid_triangle_edges L h).1) =
-      generator L φ v root hv (edgeShape L a c (valid_triangle_edges L h).2.1) :=
-    Subtype.ext he.2
-  constructor
-  · rw [lowerTriangleWord_chain L _ _ _ _ h₁,triangle_boundary_low L φ v root hv]
-  · rw [upperTriangleWord_chain L _ _ _ _ h₁ h₂,
-      triangle_boundary_high L φ v root hv comm]
-
-def triangleWordList : (w : List A) → SimplicialChains.Valid L w → ℤ → FreeGroup (Shape L 2)
-  | [a,b,c],h,n =>
-    let e := valid_triangle_edges L h
-    if n = -1 then
-      lowerTriangleWord L (edgeShape L a b e.1) (edgeShape L a c e.2.1) (edgeShape L b c e.2.2)
-    else upperTriangleWord L (edgeShape L a b e.1) (edgeShape L a c e.2.1) (edgeShape L b c e.2.2)
-  | _,_,_ => 1
-
-def triangleWord (s : Shape L 3) : FreeGroup (Shape L 2) :=
-  triangleWordList L s.val.1 s.property.1 s.val.2
-
-theorem shape_three_form (s : Shape L 3) :
-    ∃ (a b c : A) (h : SimplicialChains.Valid L [a,b,c]) (n : ℤ) (hn : n = -1 ∨ n = -2),
-      s = triangleShape L a b c h n hn := by
-  obtain ⟨a,b,c,hw⟩ := List.length_eq_three.mp s.property.2.1
-  have hn : s.val.2 = -1 ∨ s.val.2 = -2 := by
-    have h₁ := s.property.2.2.1
-    have h₂ := s.property.2.2.2
-    norm_num at h₂
-    omega
-  refine ⟨a,b,c,hw ▸ s.property.1,s.val.2,hn,?_⟩
-  exact Subtype.ext (Prod.ext hw rfl)
-
-include comm in
-theorem triangleWord_trivial (s : Shape L 3) :
-    wordValue (generator L φ v root hv) (triangleWord L s) = 1 := by
-  obtain ⟨a,b,c,h,n,hn,rfl⟩ := shape_three_form L s
-  rcases hn with rfl | rfl
-  · simpa [triangleWord,triangleWordList,triangleShape] using
-      (triangle_words_trivial L φ v root hv comm a b c h).1
-  · simpa [triangleWord,triangleWordList,triangleShape] using
-      (triangle_words_trivial L φ v root hv comm a b c h).2
-
-include comm in
-theorem triangleWord_chain (s : Shape L 3) (g : φ.ker) :
-    pathChain (generator L φ v root hv) (triangleWord L s) g =
-      triangleBoundary L φ v root hv (Finsupp.single (g,s) 1) := by
-  obtain ⟨a,b,c,h,n,hn,rfl⟩ := shape_three_form L s
-  rcases hn with rfl | rfl
-  · simpa [triangleWord,triangleWordList,triangleShape] using
-      (triangle_words_chain L φ v root hv comm g a b c h).1
-  · simpa [triangleWord,triangleWordList,triangleShape] using
-      (triangle_words_chain L φ v root hv comm g a b c h).2
-
-end EilenbergGanea.LevelBasis
-namespace EilenbergGanea.LevelBasis
-open CubicalChains CubicalChains.HeightFiltration
-variable {G A : Type} [Group G] [LinearOrder A]
-variable (L : SimpleGraph A) (φ : G →* Multiplicative ℤ) (v : A → G) (root : A)
-variable (hv : ∀ a, φ (v a) = Multiplicative.ofAdd (1 : ℤ))
-variable (comm : ∀ a b, L.Adj a b → Commute (v a) (v b))
-
-include comm in
-theorem triangleBoundary_eq_relatorChainMap :
-    triangleBoundary L φ v root hv =
-      relatorChainMap (generator L φ v root hv) (triangleWord L) := by
-  apply Finsupp.lhom_ext'
-  intro p
-  apply LinearMap.ext
-  intro n
-  simp only [LinearMap.coe_comp,Function.comp_apply,Finsupp.lsingle_apply,
-    relatorChainMap,Finsupp.linearCombination_single]
-  have he : Finsupp.single p n = n • Finsupp.single p (1 : ℤ) := by simp
-  rw [he,map_smul,← triangleWord_chain L φ v root hv comm p.2 p.1]
-
-variable (K : SimplicialChains.Contraction L)
-variable (exactTotal : ∀ {k : ℕ}, 0 < k → ∀ c, c ∈ total (G := G) L k → boundary v c = 0 →
-      ∃ b ∈ total (G := G) L (k+1), boundary v b = c)
-variable (dim : ∀ w, SimplicialChains.Valid L w → w.length ≤ 3)
-
-noncomputable def triangleBasis : (φ.ker × Shape L 3 →₀ ℤ) ≃ₗ[ℤ]
-    cycles (generator L φ v root hv) :=
-  LinearEquiv.ofBijective
-    ((triangleBoundary L φ v root hv).codRestrict (cycles (generator L φ v root hv))
-      (triangleBoundary_cycle L φ v root hv comm))
-    ⟨fun c d h => triangleBoundary_injective L φ v root hv comm K exactTotal dim
-      (congrArg Subtype.val h),fun c => by
-        have hc : c.val ∈ LinearMap.range (triangleBoundary L φ v root hv) := by
-          rw [triangleBoundary_range L φ v root hv comm K exactTotal]
-          exact c.property
-        obtain ⟨b,hb⟩ := hc
-        exact ⟨b,Subtype.ext hb⟩⟩
-
-theorem triangleBasis_value (c : φ.ker × Shape L 3 →₀ ℤ) :
-    (triangleBasis L φ v root hv comm K exactTotal dim c : φ.ker × Shape L 2 →₀ ℤ) =
-      relatorChainMap (generator L φ v root hv) (triangleWord L) c := by
-  change triangleBoundary L φ v root hv c = _
-  rw [triangleBoundary_eq_relatorChainMap L φ v root hv comm]
-
-end EilenbergGanea.LevelBasis
-namespace EilenbergGanea.LevelBasis
-open LevelTransport
-variable {V : Type} [LinearOrder V] (L : SimpleGraph V) (root : V) (N : ℕ)
-
-noncomputable def levelEquiv : (height L).ker ≃ NegativeLevel L N where
-  toFun := fun g => ⟨(artinGenerator L root)⁻¹ ^ N * g.val,by
-    simp only [map_mul,map_pow,map_inv,height_generator]
-    rw [show height L g.val = 1 from g.property,mul_one]
-    change N • (-1 : ℤ) = -(N : ℤ)
-    simp⟩
-  invFun := fun g => ⟨artinGenerator L root ^ N * g.val,by
-    change (height L (artinGenerator L root ^ N * g.val)).toAdd = 0
-    simp only [map_mul,map_pow,height_generator]
-    change N • (1 : ℤ) + (height L g.val).toAdd = 0
-    rw [g.property]
-    simp⟩
-  left_inv := fun g => Subtype.ext (by dsimp; group)
-  right_inv := fun g => Subtype.ext (by dsimp; group)
-
-def basedLabel (a : NegativeLevel L N → V → V → SU2)
-    (g : (height L).ker) (s : Shape L 2) : SU2 :=
-  a (levelEquiv L root N g) (s.val.1.tail.headD root) (s.val.1.headD root)
-
-@[simp] theorem basedLabel_edge (a : NegativeLevel L N → V → V → SU2)
-    (g : (height L).ker) (q r : V) (h : SimplicialChains.Valid L [q,r]) :
-    basedLabel L root N a g (edgeShape L q r h) = a (levelEquiv L root N g) r q := rfl
-
-theorem levelEquiv_edge (g : (height L).ker) (q r : V)
-    (h : SimplicialChains.Valid L [q,r]) :
-    levelEquiv L root N
-        (g * generator L (height L) (artinGenerator L) root (height_generator L) (edgeShape L q r h)) =
-      edgeVertex L N (levelEquiv L root N g) r q := by
-  have ha : L.Adj q r := by simpa [SimplicialChains.Valid] using h.2
-  apply Subtype.ext
-  simp only [levelEquiv,Equiv.coe_fn_mk,Subgroup.coe_mul,generator_edgeShape,edgeVertex]
-  rw [mul_assoc, (adjacent_generators_commute L ha).inv_left.eq]
-  group
-
-theorem shape_two_presentation (s : Shape L 2) :
-    ∃ (q r : V) (h : SimplicialChains.Valid L [q,r]), s = edgeShape L q r h := by
-  obtain ⟨q,r,hw⟩ := List.length_eq_two.mp s.property.2.1
-  have hn : s.val.2 = -1 := by
-    have h₁ := s.property.2.2.1
-    have h₂ := s.property.2.2.2
-    norm_num at h₂
-    omega
-  refine ⟨q,r,hw ▸ s.property.1,?_⟩
-  exact Subtype.ext (Prod.ext hw hn)
-
-theorem levelEquiv_edge_inv (g : (height L).ker) (q r : V)
-    (h : SimplicialChains.Valid L [q,r]) :
-    levelEquiv L root N
-        (g * (generator L (height L) (artinGenerator L) root (height_generator L) (edgeShape L q r h))⁻¹) =
-      edgeVertex L N (levelEquiv L root N g) q r := by
-  have he := levelEquiv_edge L root N
-    (g * (generator L (height L) (artinGenerator L) root (height_generator L) (edgeShape L q r h))⁻¹)
-    q r h
-  simp only [inv_mul_cancel_right] at he
-  simpa only [edgeVertex_reverse] using congrArg (fun x => edgeVertex L N x q r) he.symm
-
-variable (a : NegativeLevel L N → V → V → SU2)
-variable (rev : ∀ g q r, a (edgeVertex L N g q r) r q = (a g q r)⁻¹)
-include rev in
-theorem basedLabel_inv_edge (g : (height L).ker) (q r : V)
-    (h : SimplicialChains.Valid L [q,r]) :
-    (basedLabel L root N a
-      (g * (generator L (height L) (artinGenerator L) root (height_generator L) (edgeShape L q r h))⁻¹)
-      (edgeShape L q r h))⁻¹ = a (levelEquiv L root N g) q r := by
-  rw [basedLabel_edge,levelEquiv_edge_inv,rev,inv_inv]
-
-def orientedWord (q r : V) (h : L.Adj q r) : FreeGroup (Shape L 2) :=
-  if hlt : q < r then
-    (FreeGroup.of (edgeShape L q r (by simpa [SimplicialChains.Valid] using And.intro hlt h)))⁻¹
-  else
-    FreeGroup.of (edgeShape L r q (by
-      have hlt' : r < q := lt_of_le_of_ne (le_of_not_gt hlt) (Ne.symm (L.ne_of_adj h))
-      simpa [SimplicialChains.Valid] using And.intro hlt' h.symm))
-
-theorem orientedWord_value (q r : V) (h : L.Adj q r) :
-    (wordValue (generator L (height L) (artinGenerator L) root (height_generator L))
-      (orientedWord L q r h) : ArtinGroup L) =
-        artinGenerator L q * (artinGenerator L r)⁻¹ := by
-  unfold orientedWord
-  split_ifs
-  · simp only [map_inv,wordValue,FreeGroup.lift_apply_of,Subgroup.coe_inv,generator_edgeShape,
-      mul_inv_rev,inv_inv]
-    exact (adjacent_generators_commute L h).inv_right.symm.eq
-  · simp only [wordValue,FreeGroup.lift_apply_of,generator_edgeShape]
-    exact (adjacent_generators_commute L h).inv_right.symm.eq
-
-theorem levelEquiv_orientedWord (g : (height L).ker) (q r : V) (h : L.Adj q r) :
-    levelEquiv L root N
-      (g * wordValue (generator L (height L) (artinGenerator L) root (height_generator L))
-        (orientedWord L q r h)) = edgeVertex L N (levelEquiv L root N g) q r := by
-  apply Subtype.ext
-  simp only [levelEquiv,Equiv.coe_fn_mk,Subgroup.coe_mul,orientedWord_value,edgeVertex,mul_assoc]
-
-include rev in
-theorem orientedWord_product (g : (height L).ker) (q r : V) (h : L.Adj q r) :
-    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
-      (basedLabel L root N a) (orientedWord L q r h) g = a (levelEquiv L root N g) q r := by
-  unfold orientedWord
-  split_ifs
-  · simp only [pathProduct_inv,pathProduct_of,wordValue,FreeGroup.lift_apply_of]
-    exact basedLabel_inv_edge L root N a rev g q r _
-  · exact pathProduct_of _ _ _ _
-
-
-include rev in
-theorem inverse_edgeWord_product (g : (height L).ker) (q r : V)
-    (h : SimplicialChains.Valid L [q,r]) :
-    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
-      (basedLabel L root N a) (FreeGroup.of (edgeShape L q r h))⁻¹ g =
-      a (levelEquiv L root N g) q r := by
-  simp only [pathProduct_inv,pathProduct_of,wordValue,FreeGroup.lift_apply_of]
-  exact basedLabel_inv_edge L root N a rev g q r h
-
-variable (tri : ∀ g₀ g₁ g₂ q₀ r₀ q₁ r₁ q₂ r₂,
-    edgeVertex L N g₀ q₀ r₀ = g₁ → edgeVertex L N g₁ q₁ r₁ = g₂ →
-      edgeVertex L N g₂ q₂ r₂ = g₀ → a g₀ q₀ r₀ * a g₁ q₁ r₁ * a g₂ q₂ r₂ = 1)
-
-include rev tri in
-theorem lowerTriangleWord_product (g : (height L).ker) (q r s : V)
-    (h : SimplicialChains.Valid L [q,r,s]) :
-    let e := valid_triangle_edges L h
-    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
-      (basedLabel L root N a)
-      (lowerTriangleWord L (edgeShape L q r e.1) (edgeShape L q s e.2.1)
-        (edgeShape L r s e.2.2)) g = 1 := by
-  let gen := generator L (height L) (artinGenerator L) root (height_generator L)
-  let e := valid_triangle_edges L h
-  let ab := edgeShape L q r e.1
-  let ac := edgeShape L q s e.2.1
-  let bc := edgeShape L r s e.2.2
-  have he : gen ab * gen bc = gen ac :=
-    Subtype.ext (edge_values_compose L (artinGenerator L)
-      (fun _ _ hh => adjacent_generators_commute L hh) q r s h).1
-  have hc : gen ac * (gen bc)⁻¹ = gen ab := by rw [← he]; group
-  have hc' : g * gen ac * (gen bc)⁻¹ = g * gen ab := by rw [mul_assoc,hc]
-  have ha : g * gen ab * (gen ab)⁻¹ = g := by group
-  have T := tri (levelEquiv L root N g) (levelEquiv L root N (g * gen ac))
-    (levelEquiv L root N (g * gen ab)) s q r s q r
-    (levelEquiv_edge L root N g q s e.2.1).symm
-    (by rw [← levelEquiv_edge_inv L root N (g * gen ac) r s e.2.2]; exact congrArg _ hc')
-    (by rw [← levelEquiv_edge_inv L root N (g * gen ab) q r e.1]; exact congrArg _ ha)
-  change pathProduct gen (basedLabel L root N a) (lowerTriangleWord L ab ac bc) g = 1
-  simp only [lowerTriangleWord,pathProduct_mul,pathProduct_of,map_mul,map_inv,
-    wordValue,FreeGroup.lift_apply_of]
-  rw [hc]
-  rw [inverse_edgeWord_product L root N a rev _ r s e.2.2,
-    inverse_edgeWord_product L root N a rev _ q r e.1]
-  exact T
-
-include rev tri in
-theorem upperInnerWord_product (g : (height L).ker) (q r s : V)
-    (h : SimplicialChains.Valid L [q,r,s]) :
-    let e := valid_triangle_edges L h
-    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
-      (basedLabel L root N a)
-      (FreeGroup.of (edgeShape L r s e.2.2) * FreeGroup.of (edgeShape L q r e.1) *
-        (FreeGroup.of (edgeShape L q s e.2.1))⁻¹) g = 1 := by
-  let gen := generator L (height L) (artinGenerator L) root (height_generator L)
-  let e := valid_triangle_edges L h
-  let ab := edgeShape L q r e.1
-  let ac := edgeShape L q s e.2.1
-  let bc := edgeShape L r s e.2.2
-  have he : gen bc * gen ab = gen ac :=
-    Subtype.ext (edge_values_compose L (artinGenerator L)
-      (fun _ _ hh => adjacent_generators_commute L hh) q r s h).2
-  have hc : g * gen bc * gen ab = g * gen ac := by rw [mul_assoc,he]
-  have ha : g * gen ac * (gen ac)⁻¹ = g := by group
-  have T := tri (levelEquiv L root N g) (levelEquiv L root N (g * gen bc))
-    (levelEquiv L root N (g * gen ac)) s r r q q s
-    (levelEquiv_edge L root N g r s e.2.2).symm
-    (by rw [← levelEquiv_edge L root N (g * gen bc) q r e.1]; exact congrArg _ hc)
-    (by rw [← levelEquiv_edge_inv L root N (g * gen ac) q s e.2.1]; exact congrArg _ ha)
-  change pathProduct gen (basedLabel L root N a)
-    (FreeGroup.of bc * FreeGroup.of ab * (FreeGroup.of ac)⁻¹) g = 1
-  simp only [pathProduct_mul,pathProduct_of,map_mul,wordValue,FreeGroup.lift_apply_of]
-  rw [he,inverse_edgeWord_product L root N a rev _ q s e.2.1]
-  exact T
-
-include rev tri in
-theorem upperTriangleWord_product (g : (height L).ker) (q r s : V)
-    (h : SimplicialChains.Valid L [q,r,s]) :
-    let e := valid_triangle_edges L h
-    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
-      (basedLabel L root N a)
-      (upperTriangleWord L (edgeShape L q r e.1) (edgeShape L q s e.2.1)
-        (edgeShape L r s e.2.2)) g = 1 := by
-  let gen := generator L (height L) (artinGenerator L) root (height_generator L)
-  let e := valid_triangle_edges L h
-  let ab := edgeShape L q r e.1
-  let ac := edgeShape L q s e.2.1
-  let bc := edgeShape L r s e.2.2
-  have he : gen bc * gen ab = gen ac :=
-    Subtype.ext (edge_values_compose L (artinGenerator L)
-      (fun _ _ hh => adjacent_generators_commute L hh) q r s h).2
-  have hr : wordValue gen (FreeGroup.of bc * FreeGroup.of ab * (FreeGroup.of ac)⁻¹) = 1 := by
-    simp only [map_mul,map_inv,wordValue,FreeGroup.lift_apply_of,he,mul_inv_cancel]
-  change pathProduct gen (basedLabel L root N a) (upperTriangleWord L ab ac bc) g = 1
-  rw [upperTriangleWord,pathProduct_conjugate_loop gen _ _ _ hr]
-  rw [upperInnerWord_product L root N a rev tri _ q r s h]
-  group
-
-include rev tri in
-theorem triangleWord_product (s : Shape L 3) (g : (height L).ker) :
-    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
-      (basedLabel L root N a) (triangleWord L s) g = 1 := by
-  obtain ⟨q,r,t,h,n,hn,rfl⟩ := shape_three_form L s
-  rcases hn with rfl | rfl
-  · simpa [triangleWord,triangleWordList,triangleShape] using
-      lowerTriangleWord_product L root N a rev tri g q r t h
-  · simpa [triangleWord,triangleWordList,triangleShape] using
-      upperTriangleWord_product L root N a rev tri g q r t h
-
-
-variable {L N}
-def walkWord {g k : NegativeLevel L N} : Walk L N g k → FreeGroup (Shape L 2)
-  | .nil _ => 1
-  | .cons q r h p => orientedWord L q r h * walkWord p
-
- theorem walkWord_endpoint {g k : NegativeLevel L N} (p : Walk L N g k)
-    (x : (height L).ker) (hx : levelEquiv L root N x = g) :
-    levelEquiv L root N
-      (x * wordValue (generator L (height L) (artinGenerator L) root (height_generator L))
-        (walkWord p)) = k := by
-  induction p generalizing x with
-  | nil g => simpa [walkWord] using hx
-  | @cons g k q r h p ih =>
-    simp only [walkWord,map_mul,← mul_assoc]
-    exact ih _ ((levelEquiv_orientedWord L root N x q r h).trans
-      (congrArg (fun t => edgeVertex L N t q r) hx))
-
-include rev in
- theorem walkWord_product {g k : NegativeLevel L N} (p : Walk L N g k)
-    (x : (height L).ker) (hx : levelEquiv L root N x = g) :
-    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
-      (basedLabel L root N a) (walkWord p) x = p.eval a := by
-  induction p generalizing x with
-  | nil g => simp [walkWord,Walk.eval]
-  | @cons g k q r h p ih =>
-    rw [walkWord,pathProduct_mul,orientedWord_product L root N a rev x q r h]
-    rw [ih _ ((levelEquiv_orientedWord L root N x q r h).trans
-      (congrArg (fun t => edgeVertex L N t q r) hx)),hx]
-    rfl
-
-include rev in
- theorem nontrivial_word_of_walk {g : NegativeLevel L N} (p : Walk L N g g)
-    (hp : p.eval a ≠ 1) :
-    ∃ (w : FreeGroup (Shape L 2)) (x : (height L).ker),
-      wordValue (generator L (height L) (artinGenerator L) root (height_generator L)) w = 1 ∧
-      pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
-        (basedLabel L root N a) w x ≠ 1 := by
-  let x := (levelEquiv L root N).symm g
-  have hx : levelEquiv L root N x = g := (levelEquiv L root N).apply_symm_apply g
-  refine ⟨walkWord p,x,?_,?_⟩
-  · have he := walkWord_endpoint root p x hx
-    have he' := (levelEquiv L root N).injective (he.trans hx.symm)
-    exact mul_left_cancel (he'.trans (mul_one x).symm)
-  · rwa [walkWord_product root a rev p x hx]
-
-end EilenbergGanea.LevelBasis
-
-namespace EilenbergGanea.LevelBasis
-open LevelTransport
-variable {V : Type} [LinearOrder V] [Fintype V] [Nonempty V] (L : SimpleGraph V)
-variable [Group.ResiduallyFinite (height L).ker]
-variable (root : V) (hconn : L.Preconnected)
-variable (K : SimplicialChains.Contraction L)
-variable (dim : ∀ w, SimplicialChains.Valid L w → w.length ≤ 3)
-
-include root hconn K dim in
-/-- The source obstruction's algebraic endpoint, with no cardinal bound on
-an alleged presentation. Its geometric premise must still be derived from an
-ordinary two-dimensional classifying CW space. -/
-theorem no_boundary_basis_presentation
-    (small : ∀ ε : ℝ, 0 < ε →
-      ∃ N : ℕ, 1 < N ∧ ∃ a : NegativeLevel L N → V → V → SU2,
-      (∀ g q r, a (edgeVertex L N g q r) r q = (a g q r)⁻¹) ∧
-      (∀ g q r, dist (a g q r) 1 < ε) ∧
-      (∀ g₀ g₁ g₂ q₀ r₀ q₁ r₁ q₂ r₂,
-        edgeVertex L N g₀ q₀ r₀ = g₁ →
-        edgeVertex L N g₁ q₁ r₁ = g₂ →
-        edgeVertex L N g₂ q₂ r₂ = g₀ →
-        a g₀ q₀ r₀ * a g₁ q₁ r₁ * a g₂ q₂ r₂ = 1) ∧
-      ∃ (g : NegativeLevel L N) (p : Walk L N g g), p.eval a ≠ 1)
-    {A R : Type*} (a₀ : A → (height L).ker)
-    (ha₀ : Function.Surjective (wordValue a₀))
-    (r₀ : R → FreeGroup A) (hr₀ : ∀ i, wordValue a₀ (r₀ i) = 1)
-    (hp₀ : (wordValue a₀).ker = Subgroup.normalClosure (Set.range r₀))
-    (E : BasedChains (height L).ker R ≃ₗ[ℤ] cycles a₀)
-    (hE : ∀ c, (E c : BasedChains (height L).ker A) = relatorChainMap a₀ r₀ c) : False := by
-  let gen := generator L (height L) (artinGenerator L) root (height_generator L)
-  let comm := fun q r h => @adjacent_generators_commute V L q r h
-  let B := triangleBasis L (height L) (artinGenerator L) root (height_generator L)
-    comm K (artin_total_cycle_bounds L) dim
-  obtain ⟨η,hη,hηa⟩ := boundary_basis_small_transport gen
-    (generator_surjective L hconn root) (triangleWord L)
-    (triangleWord_trivial L (height L) (artinGenerator L) root (height_generator L) comm)
-    B (triangleBasis_value L (height L) (artinGenerator L) root (height_generator L)
-      comm K (artin_total_cycle_bounds L) dim) a₀ ha₀ r₀ hr₀ hp₀ E hE
-  obtain ⟨N,_,a,ha,hb,ht,g,p,hp⟩ := small η hη
-  obtain ⟨w,x,hw,hx⟩ := nontrivial_word_of_walk root a ha p hp
-  apply hx
-  exact hηa (basedLabel L root N a)
-    (fun x s => by
-      simpa only [basedLabel,Subtype.dist_eq,dist_eq_norm,OneMemClass.coe_one] using
-        hb (levelEquiv L root N x) (s.val.1.tail.headD root) (s.val.1.headD root))
-    (triangleWord_product L root N a ha ht) w hw x
-
-end EilenbergGanea.LevelBasis
-
-
-end
-
-end OAI
+import OAI.AlgebraicTopology.Cubical.HeightChains
+import OAI.Topology.EilenbergGanea.IntegralModules
+import OAI.Topology.EilenbergGanea.LevelTransport
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set Filter Topology MeasureTheory
+open scoped Quaternion ContDiff
+
+namespace EilenbergGanea.LevelBasis
+open CubicalChains CubicalChains.LevelExact CubicalChains.HeightFiltration
+variable {G A : Type} [Group G] [LinearOrder A]
+variable (L : SimpleGraph A) (φ : G →* Multiplicative ℤ) (v : A → G) (root : A)
+def ht (g : G) : ℤ := (φ g).toAdd
+@[simp] theorem ht_mul (g q : G) : ht φ (g*q) = ht φ g + ht φ q := by simp [ht]
+@[simp] theorem ht_one : ht φ 1 = 0 := by simp [ht]
+@[simp] theorem ht_inv (g : G) : ht φ g⁻¹ = -ht φ g := by simp [ht]
+@[simp] theorem ht_zpow (g : G) (n : ℤ) : ht φ (g^n) = n * ht φ g := by simp [ht]
+@[simp] theorem ht_ker (g : φ.ker) : ht φ g = 0 := by simp [ht,MonoidHom.mem_ker.mp g.property]
+variable (hv : ∀ a, φ (v a) = Multiplicative.ofAdd (1 : ℤ))
+include hv in
+omit [LinearOrder A] in
+@[simp] theorem ht_generator (a : A) : ht φ (v a) = 1 := by simp [ht,hv]
+include hv in
+omit [LinearOrder A] in
+theorem ht_step (g : G) (a : A) : ht φ (g * v a) = ht φ g + 1 := by simp [ht_generator φ v hv]
+
+def Shape (k : ℕ) := {p : List A × ℤ // SimplicialChains.Valid L p.1 ∧ p.1.length = k ∧
+  p.2 < 0 ∧ 0 < p.2 + (k : ℤ)}
+def basisIndex (k : ℕ) (p : φ.ker × Shape L k) : G × List A :=
+  (p.1.val * (v (p.2.val.1.headD root)) ^ p.2.val.2,p.2.val.1)
+
+include hv in
+theorem basisIndex_height (k : ℕ) (p : φ.ker × Shape L k) :
+    ht φ (basisIndex L φ v root k p).1 = p.2.val.2 := by
+  simp only [basisIndex,ht_mul,ht_ker,ht_zpow,ht_generator φ v hv,mul_one,zero_add]
+
+include hv in
+theorem basisIndex_injective (k : ℕ) : Function.Injective (basisIndex L φ v root k) := by
+  intro p q he
+  have hw : p.2.val.1 = q.2.val.1 := congrArg Prod.snd he
+  have hh := congrArg (fun x => ht φ x.1) he
+  rw [basisIndex_height L φ v root hv,basisIndex_height L φ v root hv] at hh
+  have hs : p.2 = q.2 := Subtype.ext (Prod.ext hw hh)
+  have hg := congrArg Prod.fst he
+  simp only [basisIndex,hs] at hg
+  exact Prod.ext (Subtype.ext (mul_right_cancel hg)) hs
+
+include hv in
+theorem basisIndex_mem (k : ℕ) (p : φ.ker × Shape L k) :
+    SimplicialChains.Valid L (basisIndex L φ v root k p).2 ∧
+      (basisIndex L φ v root k p).2.length = k ∧
+      ht φ (basisIndex L φ v root k p).1 < 0 ∧
+      0 < ht φ (basisIndex L φ v root k p).1 + (k : ℤ) := by
+  rw [basisIndex_height L φ v root hv]
+  exact p.2.property
+
+include hv in
+theorem exists_basisIndex {k : ℕ} (p : G × List A)
+    (hp : SimplicialChains.Valid L p.2 ∧ p.2.length = k ∧ ht φ p.1 < 0 ∧ 0 < ht φ p.1 + (k : ℤ)) :
+    ∃ q : φ.ker × Shape L k, basisIndex L φ v root k q = p := by
+  let s : Shape L k := ⟨(p.2,ht φ p.1),hp⟩
+  let g : φ.ker := ⟨p.1 * (v (p.2.headD root)) ^ (-ht φ p.1),by
+    apply (Multiplicative.toAdd.injective)
+    change ht φ (p.1 * (v (p.2.headD root)) ^ (-ht φ p.1)) = 0
+    simp only [ht_mul,ht_zpow,ht_generator φ v hv,mul_one,add_neg_cancel]⟩
+  refine ⟨(g,s),?_⟩
+  apply Prod.ext
+  · change (p.1 * (v (p.2.headD root)) ^ (-ht φ p.1)) * (v (p.2.headD root)) ^ (ht φ p.1) = p.1
+    rw [mul_assoc,← zpow_add,neg_add_cancel,zpow_zero,mul_one]
+  · rfl
+
+def includeBasis (k : ℕ) : (φ.ker × Shape L k →₀ ℤ) →ₗ[ℤ] CubicalChains.Chains G A :=
+  Finsupp.lmapDomain ℤ ℤ (basisIndex L φ v root k)
+@[simp] theorem includeBasis_single (k : ℕ) (p : φ.ker × Shape L k) (n : ℤ) :
+    includeBasis L φ v root k (Finsupp.single p n) = Finsupp.single (basisIndex L φ v root k p) n := by
+  simp [includeBasis]
+include hv in
+theorem includeBasis_injective (k : ℕ) : Function.Injective (includeBasis L φ v root k) :=
+  Finsupp.mapDomain_injective (basisIndex_injective L φ v root hv k)
+
+include hv in
+theorem includeBasis_range (k : ℕ) : LinearMap.range (includeBasis L φ v root k) = crossing L (ht φ) k := by
+  rw [includeBasis,Finsupp.range_lmapDomain,crossing,Finsupp.supported_eq_span_single]
+  congr 1
+  ext c
+  constructor
+  · rintro ⟨p,rfl⟩
+    exact ⟨basisIndex L φ v root k p,basisIndex_mem L φ v root hv k p,rfl⟩
+  · rintro ⟨p,hp,rfl⟩
+    obtain ⟨q,rfl⟩ := exists_basisIndex L φ v root hv p hp
+    exact ⟨q,rfl⟩
+
+variable (comm : ∀ a b, L.Adj a b → Commute (v a) (v b))
+include hv in
+def projectBasis (k : ℕ) : CubicalChains.Chains G A →ₗ[ℤ] (φ.ker × Shape L k →₀ ℤ) :=
+  Finsupp.lcomapDomain (basisIndex L φ v root k) (basisIndex_injective L φ v root hv k)
+
+@[simp] theorem project_include (k : ℕ) (c : φ.ker × Shape L k →₀ ℤ) :
+    projectBasis L φ v root hv k (includeBasis L φ v root k c) = c :=
+  Finsupp.leftInverse_lcomapDomain_mapDomain _ _ c
+
+include hv in
+theorem include_project {k : ℕ} {c : CubicalChains.Chains G A} (hc : c ∈ crossing L (ht φ) k) :
+    includeBasis L φ v root k (projectBasis L φ v root hv k c) = c := by
+  rw [← includeBasis_range L φ v root hv k] at hc
+  obtain ⟨d,rfl⟩ := hc
+  rw [project_include]
+
+include hv in
+theorem includeBasis_mem (k : ℕ) (c : φ.ker × Shape L k →₀ ℤ) :
+    includeBasis L φ v root k c ∈ crossing L (ht φ) k := by
+  rw [← includeBasis_range L φ v root hv k]
+  exact ⟨c,rfl⟩
+
+/-- The level edge parameter has precisely two directions and initial height -1. -/
+theorem shape_two_form (s : Shape L 2) : ∃ a b, s.val = ([a,b],-1) := by
+  obtain ⟨a,b,hab⟩ := List.length_eq_two.mp s.property.2.1
+  refine ⟨a,b,Prod.ext hab ?_⟩
+  have hn := s.property.2.2.1
+  have hm := s.property.2.2.2
+  norm_num at hm
+  omega
+
+def generator (s : Shape L 2) : φ.ker :=
+  ⟨(v (s.val.1.headD root))⁻¹ * v (s.val.1.tail.headD root),by simp [hv]⟩
+
+def includeVertices : (φ.ker →₀ ℤ) →ₗ[ℤ] CubicalChains.Chains G A :=
+  Finsupp.lmapDomain ℤ ℤ (fun g => (g.val,[]))
+omit [LinearOrder A] in
+@[simp] theorem includeVertices_single (g : φ.ker) (n : ℤ) :
+    includeVertices (A := A) φ (Finsupp.single g n) = Finsupp.single (g.val,[]) n := by simp [includeVertices]
+omit [LinearOrder A] in
+theorem includeVertices_injective : Function.Injective (includeVertices (A := A) φ) := by
+  apply Finsupp.mapDomain_injective
+  intro p q he
+  exact Subtype.ext (congrArg Prod.fst he)
+
+include hv in
+omit [LinearOrder A] in
+theorem boundary_square (g : G) (a b : A) (hab : Commute (v a) (v b))
+    (hg : ht φ g = -1) :
+    levelBoundary v (ht φ) (Finsupp.single (g,[a,b]) 1) =
+      Finsupp.single (g*v b,[]) 1 - Finsupp.single (g*v a,[]) 1 := by
+  have hga : ht φ (g * v a) = 0 := by rw [ht_step φ v hv,hg]; omega
+  have hgb : ht φ (g * v b) = 0 := by rw [ht_step φ v hv,hg]; omega
+  simp only [levelBoundary_apply,boundary_single,one_smul,boundaryTerm_cons,boundaryTerm_nil,
+    map_zero,sub_zero,map_sub,front_single,plus,select_single,hga,hgb,hg]
+  norm_num
+  rw [mul_assoc,hab.eq,← mul_assoc]
+  abel
+
+
+include hv comm in
+theorem levelBoundary_include_single (p : φ.ker × Shape L 2) :
+    levelBoundary v (ht φ) (includeBasis L φ v root 2 (Finsupp.single p 1)) =
+      includeVertices (A := A) φ (edgeBoundary (generator L φ v root hv) (Finsupp.single p 1)) := by
+  rcases p with ⟨g,⟨⟨w,n⟩,hs⟩⟩
+  obtain ⟨a,b,hw⟩ := List.length_eq_two.mp hs.2.1
+  change w = [a,b] at hw
+  subst w
+  have hn : n = -1 := by have h₁ := hs.2.2.1; have h₂ := hs.2.2.2; norm_num at h₂; omega
+  subst n
+  have hab : L.Adj a b := (List.pairwise_cons.mp hs.1.2).1 b (by simp)
+  rw [includeBasis_single]
+  change levelBoundary v (ht φ) (Finsupp.single (g.val * (v a)^(-1 : ℤ),[a,b]) 1) = _
+  rw [boundary_square φ v hv _ a b (comm a b hab) (by simp [ht_generator φ v hv])]
+  simp only [edgeBoundary,Finsupp.linearCombination_single]
+  simp only [zpow_neg_one,one_smul,map_sub,includeVertices_single,
+    Subgroup.coe_mul,generator,List.headD_cons,List.tail_cons,mul_assoc,inv_mul_cancel,mul_one]
+
+include hv comm in
+theorem levelBoundary_include (c : φ.ker × Shape L 2 →₀ ℤ) :
+    levelBoundary v (ht φ) (includeBasis L φ v root 2 c) =
+      includeVertices (A := A) φ (edgeBoundary (generator L φ v root hv) c) := by
+  induction c using Finsupp.induction_linear with
+  | zero => simp
+  | add c d hc hd => simp only [map_add,hc,hd]
+  | single p n =>
+    have he : Finsupp.single p n = n • Finsupp.single p (1 : ℤ) := by simp
+    rw [he,map_smul,map_smul,map_smul,map_smul,levelBoundary_include_single L φ v root hv comm]
+
+/-- The literal based integral boundary of the crossing three-cubes. -/
+def triangleBoundary : (φ.ker × Shape L 3 →₀ ℤ) →ₗ[ℤ] (φ.ker × Shape L 2 →₀ ℤ) :=
+  (projectBasis L φ v root hv 2).comp ((crossPart (ht φ)).comp ((boundary v).comp (includeBasis L φ v root 3)))
+
+include hv in
+theorem triangleBoundary_inclusion (c : φ.ker × Shape L 3 →₀ ℤ) :
+    includeBasis L φ v root 2 (triangleBoundary L φ v root hv c) =
+      crossPart (ht φ) (boundary v (includeBasis L φ v root 3 c)) := by
+  apply include_project L φ v root hv
+  exact crossPart_mem L (ht φ) (boundary_total L v (ht φ) (ht_step φ v hv)
+    (crossing_total L (ht φ) (includeBasis_mem L φ v root hv 3 c)))
+
+include hv comm in
+theorem triangleBoundary_cycle (c : φ.ker × Shape L 3 →₀ ℤ) :
+    triangleBoundary L φ v root hv c ∈ cycles (generator L φ v root hv) := by
+  apply includeVertices_injective φ
+  rw [map_zero,← levelBoundary_include L φ v root hv comm,triangleBoundary_inclusion L φ v root hv]
+  exact levelBoundary_cross_boundary L v (ht φ) (ht_step φ v hv) comm (includeBasis_mem L φ v root hv 3 c)
+
+variable (K : SimplicialChains.Contraction L)
+variable (exactTotal : ∀ {k : ℕ}, 0 < k → ∀ c, c ∈ total (G := G) L k → boundary v c = 0 →
+      ∃ b ∈ total (G := G) L (k+1), boundary v b = c)
+variable (dim : ∀ w, SimplicialChains.Valid L w → w.length ≤ 3)
+include hv comm K exactTotal in
+theorem triangleBoundary_range : LinearMap.range (triangleBoundary L φ v root hv) =
+    cycles (generator L φ v root hv) := by
+  apply le_antisymm
+  · rintro _ ⟨c,rfl⟩
+    exact triangleBoundary_cycle L φ v root hv comm c
+  · intro c hc
+    have hz : levelBoundary v (ht φ) (includeBasis L φ v root 2 c) = 0 := by
+      rw [levelBoundary_include L φ v root hv comm,LinearMap.mem_ker.mp hc,map_zero]
+    obtain ⟨t,ht,hdt⟩ := level_cycle_bounds L v (ht φ) K (ht_step φ v hv) comm exactTotal
+      _ (includeBasis_mem L φ v root hv 2 c) hz
+    rw [← includeBasis_range L φ v root hv 3] at ht
+    obtain ⟨b,rfl⟩ := ht
+    refine ⟨b,includeBasis_injective L φ v root hv 2 ?_⟩
+    rwa [triangleBoundary_inclusion L φ v root hv]
+
+include hv comm K exactTotal dim in
+theorem triangleBoundary_injective : Function.Injective (triangleBoundary L φ v root hv) := by
+  suffices hz : ∀ c, triangleBoundary L φ v root hv c = 0 → c = 0 by
+    intro c d he
+    exact sub_eq_zero.mp (hz (c-d) (by simp only [map_sub,he,sub_self]))
+  intro c hc
+  apply includeBasis_injective L φ v root hv 3
+  rw [map_zero]
+  apply level_boundary_injective L v (ht φ) K (ht_step φ v hv) comm exactTotal dim
+    _ (includeBasis_mem L φ v root hv 3 c)
+  rw [← triangleBoundary_inclusion L φ v root hv,hc,map_zero]
+
+
+/-- Left translation on the actual ambient cube chains. -/
+def leftTranslate (g : G) : CubicalChains.Chains G A →ₗ[ℤ] CubicalChains.Chains G A :=
+  Finsupp.lmapDomain ℤ ℤ (fun p => (g*p.1,p.2))
+omit [LinearOrder A] in
+@[simp] theorem leftTranslate_single (g q : G) (w : List A) (n : ℤ) :
+    leftTranslate g (Finsupp.single (q,w) n) = Finsupp.single (g*q,w) n := by simp [leftTranslate]
+omit [LinearOrder A] in
+theorem leftTranslate_front (g : G) (a : A) (c : CubicalChains.Chains G A) :
+    leftTranslate g (CubicalChains.front a c) = CubicalChains.front a (leftTranslate g c) := by
+  induction c using Finsupp.induction_linear with
+  | zero => simp
+  | add c d hc hd => simp only [map_add,hc,hd]
+  | single p n => rcases p with ⟨q,w⟩; simp
+omit [LinearOrder A] in
+theorem leftTranslate_boundaryTerm (g q : G) (w : List A) :
+    leftTranslate g (boundaryTerm v q w) = boundaryTerm v (g*q) w := by
+  induction w with
+  | nil => simp
+  | cons a w ih => simp only [boundaryTerm_cons,map_sub,leftTranslate_single,leftTranslate_front,ih,mul_assoc]
+omit [LinearOrder A] in
+theorem leftTranslate_boundary (g : G) (c : CubicalChains.Chains G A) :
+    leftTranslate g (boundary v c) = boundary v (leftTranslate g c) := by
+  induction c using Finsupp.induction_linear with
+  | zero => simp
+  | add c d hc hd => simp only [map_add,hc,hd]
+  | single p n => rcases p with ⟨q,w⟩; simp [leftTranslate_boundaryTerm]
+omit [LinearOrder A] in
+theorem leftTranslate_crossPart (g : φ.ker) (c : CubicalChains.Chains G A) :
+    leftTranslate g.val (crossPart (ht φ) c) = crossPart (ht φ) (leftTranslate g.val c) := by
+  induction c using Finsupp.induction_linear with
+  | zero => simp
+  | add c d hc hd => simp only [map_add,hc,hd]
+  | single p n =>
+      rcases p with ⟨q,w⟩
+      simp only [crossPart,select_single,leftTranslate_single,ht_mul,ht_ker,zero_add]
+      split <;> simp_all
+
+theorem includeBasis_translate (k : ℕ) (g : φ.ker) (c : φ.ker × Shape L k →₀ ℤ) :
+    includeBasis L φ v root k (translateChains g c) = leftTranslate g.val (includeBasis L φ v root k c) := by
+  induction c using Finsupp.induction_linear with
+  | zero => simp
+  | add c d hc hd => simp only [map_add,hc,hd]
+  | single p n =>
+      rcases p with ⟨q,s⟩
+      simp only [translateChains_single,includeBasis_single,leftTranslate_single,basisIndex,
+        Subgroup.coe_mul,mul_assoc]
+
+include hv in
+theorem triangleBoundary_equivariant : ChainsEquivariant (triangleBoundary L φ v root hv) := by
+  intro g c
+  apply includeBasis_injective L φ v root hv 2
+  rw [triangleBoundary_inclusion L φ v root hv,includeBasis_translate,
+    ← leftTranslate_boundary,← leftTranslate_crossPart,includeBasis_translate,
+    triangleBoundary_inclusion L φ v root hv]
+
+end EilenbergGanea.LevelBasis
+
+namespace EilenbergGanea.LevelBasis
+variable {V : Type} [LinearOrder V] (L : SimpleGraph V)
+
+def pairShape (a b : V) (hab : a < b) (hadj : L.Adj a b) : Shape L 2 :=
+  ⟨([a,b],-1),by simp [SimplicialChains.Valid,hab,hadj]⟩
+
+@[simp] theorem generator_pair (root a b : V) (hab : a < b) (hadj : L.Adj a b) :
+    (generator L (height L) (artinGenerator L) root (height_generator L)
+      (pairShape L a b hab hadj)).val = (artinGenerator L a)⁻¹ * artinGenerator L b := rfl
+
+/-- The actual oriented crossing-square edges generate precisely the height kernel. -/
+theorem generator_surjective (hconn : L.Preconnected) (root : V) :
+    Function.Surjective (FreeGroup.lift (generator L (height L) (artinGenerator L) root (height_generator L))) := by
+  let f := FreeGroup.lift (generator L (height L) (artinGenerator L) root (height_generator L))
+  let F := (height L).ker.subtype.comp f
+  have hgen : edgeGenerated L ≤ F.range := by
+    apply (Subgroup.closure_le _).mpr
+    rintro x ⟨a,b,hadj,rfl⟩
+    have hab : a ≠ b := L.ne_of_adj hadj
+    rcases lt_or_gt_of_ne hab with hab | hab
+    · refine ⟨(FreeGroup.of (pairShape L a b hab hadj))⁻¹,?_⟩
+      change (f ((FreeGroup.of (pairShape L a b hab hadj))⁻¹)).val = _
+      simp only [f,map_inv,FreeGroup.lift_apply_of,Subgroup.coe_inv,generator_pair,mul_inv_rev,inv_inv]
+      exact (adjacent_generators_commute L hadj).inv_right.eq.symm
+    · refine ⟨FreeGroup.of (pairShape L b a hab hadj.symm),?_⟩
+      change (f (FreeGroup.of (pairShape L b a hab hadj.symm))).val = _
+      simp only [f,FreeGroup.lift_apply_of,generator_pair]
+      exact (adjacent_generators_commute L hadj).inv_right.eq.symm
+  intro g
+  have he : g.val ∈ edgeGenerated L := by rw [edgeGenerated_eq_height_kernel L hconn root]; exact g.property
+  obtain ⟨w,hw⟩ := hgen he
+  exact ⟨w,Subtype.ext hw⟩
+
+end EilenbergGanea.LevelBasis
+
+namespace EilenbergGanea.LevelBasis
+open CubicalChains CubicalChains.HeightFiltration
+variable {V : Type} [LinearOrder V] [Fintype V] [Nonempty V] (L : SimpleGraph V)
+
+/-- The exactness input to the level slicing is the actual RAAG cube-chain
+contraction, not an assumption about a classifying space. -/
+theorem artin_total_cycle_bounds {k : ℕ} (hk : 0 < k)
+    (c : Chains (ArtinGroup L) V) (hc : c ∈ total (G := ArtinGroup L) L k)
+    (hz : boundary (artinGenerator L) c = 0) :
+    ∃ b ∈ total (G := ArtinGroup L) L (k+1), boundary (artinGenerator L) b = c := by
+  have he : ∀ n, total (G := ArtinGroup L) L n = TraceWords.CubeDescents.positiveChains L n := by
+    intro n
+    simp only [total,TraceWords.CubeDescents.positiveChains,SimplicialChains.Valid,and_assoc]
+  rw [he] at hc ⊢
+  exact TraceWords.CubeDescents.positive_cycle_bounds L hk c hc hz
+
+/-- Integral cd at most two for the genuine height kernel, from a genuine
+finite augmented link contraction and the dimension of its flag complex. -/
+theorem artin_height_kernel_cd_le_two (root : V) (hconn : L.Preconnected)
+    (K : SimplicialChains.Contraction L)
+    (dim : ∀ w, SimplicialChains.Valid L w → w.length ≤ 3) :
+    cohomologicalDimension (height L).ker ≤ 2 := by
+  exact projectiveDimension_le_two_of_boundary_basis
+    (generator L (height L) (artinGenerator L) root (height_generator L))
+    (generator_surjective L hconn root)
+    (triangleBoundary L (height L) (artinGenerator L) root (height_generator L))
+    (triangleBoundary_equivariant L (height L) (artinGenerator L) root (height_generator L))
+    (triangleBoundary_injective L (height L) (artinGenerator L) root (height_generator L)
+      (fun _ _ h => adjacent_generators_commute L h) K (artin_total_cycle_bounds L) dim)
+    (triangleBoundary_range L (height L) (artinGenerator L) root (height_generator L)
+      (fun _ _ h => adjacent_generators_commute L h) K (artin_total_cycle_bounds L))
+end EilenbergGanea.LevelBasis
+namespace EilenbergGanea.LevelBasis
+open CubicalChains CubicalChains.LevelExact CubicalChains.HeightFiltration
+variable {G A : Type} [Group G] [LinearOrder A]
+variable (L : SimpleGraph A) (φ : G →* Multiplicative ℤ) (v : A → G) (root : A)
+variable (hv : ∀ a, φ (v a) = Multiplicative.ofAdd (1 : ℤ))
+variable (comm : ∀ a b, L.Adj a b → Commute (v a) (v b))
+
+instance shape_finite [Fintype A] (k : ℕ) : Finite (Shape L k) := by
+  let f : Shape L k → List.Vector A k × Fin k := fun s =>
+    (⟨s.val.1,s.property.2.1⟩,⟨(-s.val.2).toNat,by
+      have hn := s.property.2.2.1
+      have hm := s.property.2.2.2
+      omega⟩)
+  apply Finite.of_injective f
+  intro s t h
+  apply Subtype.ext
+  apply Prod.ext
+  · exact congrArg (fun p : List.Vector A k × Fin k => p.1.val) h
+  · have he := congrArg (fun p : List.Vector A k × Fin k => p.2.val) h
+    have hs := s.property.2.2.1
+    have ht := t.property.2.2.1
+    dsimp [f] at he
+    omega
+
+noncomputable instance shape_fintype [Fintype A] (k : ℕ) : Fintype (Shape L k) := Fintype.ofFinite _
+
+def edgeShape (a b : A) (hab : SimplicialChains.Valid L [a,b]) : Shape L 2 :=
+  ⟨([a,b],-1),hab,rfl,by norm_num,by norm_num⟩
+
+def triangleShape (a b c : A) (habc : SimplicialChains.Valid L [a,b,c])
+    (n : ℤ) (hn : n = -1 ∨ n = -2) : Shape L 3 :=
+  ⟨([a,b,c],n),habc,rfl,by omega,by norm_num; omega⟩
+
+@[simp] theorem generator_edgeShape (a b : A) (hab : SimplicialChains.Valid L [a,b]) :
+    (generator L φ v root hv (edgeShape L a b hab) : G) = (v a)⁻¹ * v b := rfl
+
+theorem valid_triangle_edges {a b c : A} (h : SimplicialChains.Valid L [a,b,c]) :
+    SimplicialChains.Valid L [a,b] ∧ SimplicialChains.Valid L [a,c] ∧
+      SimplicialChains.Valid L [b,c] := by
+  simp [SimplicialChains.Valid] at h ⊢
+  tauto
+
+include comm in
+theorem edge_values_compose (a b c : A) (h : SimplicialChains.Valid L [a,b,c]) :
+    ((v a)⁻¹ * v b) * ((v b)⁻¹ * v c) = (v a)⁻¹ * v c ∧
+    ((v b)⁻¹ * v c) * ((v a)⁻¹ * v b) = (v a)⁻¹ * v c := by
+  have he := valid_triangle_edges L h
+  have ha : L.Adj a b := by simpa [SimplicialChains.Valid] using he.1.2
+  have hb : L.Adj a c := by simpa [SimplicialChains.Valid] using he.2.1.2
+  have hc : L.Adj b c := by simpa [SimplicialChains.Valid] using he.2.2.2
+  constructor
+  · group
+  · calc
+      ((v b)⁻¹ * v c) * ((v a)⁻¹ * v b) = (v a)⁻¹ * ((v b)⁻¹ * v c) * v b := by
+        rw [← mul_assoc,
+          ((comm a b ha).inv_left.inv_right.mul_right (comm a c hb).inv_left).symm.eq]
+      _ = (v a)⁻¹ * v c := by
+        rw [mul_assoc,mul_assoc,← (comm b c hc).eq,inv_mul_cancel_left]
+
+def lowerTriangleWord (ab ac bc : Shape L 2) : FreeGroup (Shape L 2) :=
+  FreeGroup.of ac * (FreeGroup.of bc)⁻¹ * (FreeGroup.of ab)⁻¹
+
+def upperTriangleWord (ab ac bc : Shape L 2) : FreeGroup (Shape L 2) :=
+  FreeGroup.of ab * (FreeGroup.of bc * FreeGroup.of ab * (FreeGroup.of ac)⁻¹) *
+    (FreeGroup.of ab)⁻¹
+
+include comm in
+theorem triangle_words_trivial (a b c : A) (h : SimplicialChains.Valid L [a,b,c]) :
+    let e := valid_triangle_edges L h
+    let ab := edgeShape L a b e.1
+    let ac := edgeShape L a c e.2.1
+    let bc := edgeShape L b c e.2.2
+    wordValue (generator L φ v root hv) (lowerTriangleWord L ab ac bc) = 1 ∧
+      wordValue (generator L φ v root hv) (upperTriangleWord L ab ac bc) = 1 := by
+  dsimp only
+  have he := edge_values_compose L v comm a b c h
+  constructor <;> apply Subtype.ext
+  · simp only [lowerTriangleWord,map_mul,map_inv,wordValue,FreeGroup.lift_apply_of,
+      Subgroup.coe_mul,Subgroup.coe_inv,generator_edgeShape,OneMemClass.coe_one]
+    group
+  · simp only [upperTriangleWord,map_mul,map_inv,wordValue,FreeGroup.lift_apply_of,
+      Subgroup.coe_mul,Subgroup.coe_inv,generator_edgeShape,OneMemClass.coe_one]
+    rw [he.2,mul_inv_cancel,mul_one,mul_inv_cancel]
+
+include hv in
+theorem triangle_boundary_low (g : φ.ker) (a b c : A)
+    (h : SimplicialChains.Valid L [a,b,c]) :
+    let e := valid_triangle_edges L h
+    let ab := edgeShape L a b e.1
+    let ac := edgeShape L a c e.2.1
+    let bc := edgeShape L b c e.2.2
+    triangleBoundary L φ v root hv
+        (Finsupp.single (g,triangleShape L a b c h (-1) (Or.inl rfl)) 1) =
+      Finsupp.single (g,ac) 1 - Finsupp.single (g,ab) 1 -
+        Finsupp.single (g * generator L φ v root hv ab,bc) 1 := by
+  dsimp only
+  apply includeBasis_injective L φ v root hv 2
+  rw [triangleBoundary_inclusion L φ v root hv]
+  simp only [map_sub,includeBasis_single,basisIndex,triangleShape,edgeShape,
+    List.headD_cons,zpow_neg_one,Subgroup.coe_mul,generator,List.tail_cons]
+  simp only [boundary_single,one_smul,boundaryTerm_cons,boundaryTerm_nil,map_zero,
+    sub_zero,map_sub,front_single,crossPart,select,Finsupp.linearCombination_single,
+    ht_mul,ht_ker,ht_inv,ht_generator φ v hv,List.length_cons,List.length_nil]
+  norm_num
+  simp only [mul_assoc,mul_inv_cancel,mul_one]
+  abel
+
+include hv comm in
+theorem triangle_boundary_high (g : φ.ker) (a b c : A)
+    (h : SimplicialChains.Valid L [a,b,c]) :
+    let e := valid_triangle_edges L h
+    let ab := edgeShape L a b e.1
+    let ac := edgeShape L a c e.2.1
+    let bc := edgeShape L b c e.2.2
+    triangleBoundary L φ v root hv
+        (Finsupp.single (g,triangleShape L a b c h (-2) (Or.inr rfl)) 1) =
+      Finsupp.single (g * generator L φ v root hv ab,bc) 1 -
+        Finsupp.single (g * generator L φ v root hv ab,ac) 1 +
+        Finsupp.single (g * generator L φ v root hv ac,ab) 1 := by
+  dsimp only
+  have eh := valid_triangle_edges L h
+  have hab : L.Adj a b := by simpa [SimplicialChains.Valid] using eh.1.2
+  have hac : L.Adj a c := by simpa [SimplicialChains.Valid] using eh.2.1.2
+  apply includeBasis_injective L φ v root hv 2
+  rw [triangleBoundary_inclusion L φ v root hv]
+  simp only [map_sub,map_add,includeBasis_single,basisIndex,triangleShape,edgeShape,
+    List.headD_cons,Subgroup.coe_mul,generator,List.tail_cons]
+  simp only [boundary_single,one_smul,boundaryTerm_cons,boundaryTerm_nil,map_zero,
+    sub_zero,map_sub,front_single,crossPart,select,map_sub,Finsupp.linearCombination_single,
+    ht_mul,ht_ker,ht_zpow,ht_generator φ v hv,List.length_cons,List.length_nil]
+  norm_num
+  have hb : (v a)^(-2 : ℤ) * v b = (v a)⁻¹ * v b * (v a)⁻¹ := by
+    calc
+      (v a)^(-2 : ℤ) * v b = (v a)⁻¹ * ((v a)⁻¹ * v b) := by group
+      _ = (v a)⁻¹ * (v b * (v a)⁻¹) := congrArg ((v a)⁻¹ * ·) (comm a b hab).inv_left.eq
+      _ = _ := (mul_assoc _ _ _).symm
+  have hc : (v a)^(-2 : ℤ) * v c = (v a)⁻¹ * v c * (v a)⁻¹ := by
+    calc
+      (v a)^(-2 : ℤ) * v c = (v a)⁻¹ * ((v a)⁻¹ * v c) := by group
+      _ = (v a)⁻¹ * (v c * (v a)⁻¹) := congrArg ((v a)⁻¹ * ·) (comm a c hac).inv_left.eq
+      _ = _ := (mul_assoc _ _ _).symm
+  have ha : (v a)^(-2 : ℤ) * v a = (v a)⁻¹ := by group
+  simp only [zpow_neg,zpow_ofNat] at ha hb hc
+  simp only [mul_assoc] at hb hc
+  simp only [mul_assoc,ha,hb,hc,mul_inv_cancel,mul_one]
+  abel
+
+section WordChains
+variable (w : Shape L 2 → G) (ab ac bc : Shape L 2)
+theorem lowerTriangleWord_chain (habc : w ab * w bc = w ac) (g : G) :
+    pathChain w (lowerTriangleWord L ab ac bc) g =
+      Finsupp.single (g,ac) 1 - Finsupp.single (g,ab) 1 -
+        Finsupp.single (g * w ab,bc) 1 := by
+  simp only [lowerTriangleWord,pathChain_mul,pathChain_inv,pathChain_of,
+    map_mul,map_inv,wordValue,FreeGroup.lift_apply_of]
+  rw [← habc]
+  simp only [mul_assoc,mul_inv_cancel,mul_one]
+  abel
+
+theorem upperTriangleWord_chain (habc : w ab * w bc = w ac)
+    (hbac : w bc * w ab = w ac) (g : G) :
+    pathChain w (upperTriangleWord L ab ac bc) g =
+      Finsupp.single (g * w ab,bc) 1 - Finsupp.single (g * w ab,ac) 1 +
+        Finsupp.single (g * w ac,ab) 1 := by
+  have hr : wordValue w (FreeGroup.of bc * FreeGroup.of ab * (FreeGroup.of ac)⁻¹) = 1 := by
+    simp only [wordValue,map_mul,map_inv,FreeGroup.lift_apply_of,hbac,mul_inv_cancel]
+  rw [upperTriangleWord,pathChain_conjugate w _ _ hr]
+  simp only [pathChain_mul,pathChain_inv,pathChain_of,map_mul,wordValue,
+    FreeGroup.lift_apply_of]
+  rw [hbac]
+  simp only [mul_assoc,habc,mul_inv_cancel,mul_one]
+  abel
+end WordChains
+
+include comm in
+theorem triangle_words_chain (g : φ.ker) (a b c : A) (h : SimplicialChains.Valid L [a,b,c]) :
+    let e := valid_triangle_edges L h
+    let ab := edgeShape L a b e.1
+    let ac := edgeShape L a c e.2.1
+    let bc := edgeShape L b c e.2.2
+    pathChain (generator L φ v root hv) (lowerTriangleWord L ab ac bc) g =
+      triangleBoundary L φ v root hv
+        (Finsupp.single (g,triangleShape L a b c h (-1) (Or.inl rfl)) 1) ∧
+    pathChain (generator L φ v root hv) (upperTriangleWord L ab ac bc) g =
+      triangleBoundary L φ v root hv
+        (Finsupp.single (g,triangleShape L a b c h (-2) (Or.inr rfl)) 1) := by
+  dsimp only
+  have he := edge_values_compose L v comm a b c h
+  have h₁ : generator L φ v root hv (edgeShape L a b (valid_triangle_edges L h).1) *
+      generator L φ v root hv (edgeShape L b c (valid_triangle_edges L h).2.2) =
+      generator L φ v root hv (edgeShape L a c (valid_triangle_edges L h).2.1) :=
+    Subtype.ext he.1
+  have h₂ : generator L φ v root hv (edgeShape L b c (valid_triangle_edges L h).2.2) *
+      generator L φ v root hv (edgeShape L a b (valid_triangle_edges L h).1) =
+      generator L φ v root hv (edgeShape L a c (valid_triangle_edges L h).2.1) :=
+    Subtype.ext he.2
+  constructor
+  · rw [lowerTriangleWord_chain L _ _ _ _ h₁,triangle_boundary_low L φ v root hv]
+  · rw [upperTriangleWord_chain L _ _ _ _ h₁ h₂,
+      triangle_boundary_high L φ v root hv comm]
+
+def triangleWordList : (w : List A) → SimplicialChains.Valid L w → ℤ → FreeGroup (Shape L 2)
+  | [a,b,c],h,n =>
+    let e := valid_triangle_edges L h
+    if n = -1 then
+      lowerTriangleWord L (edgeShape L a b e.1) (edgeShape L a c e.2.1) (edgeShape L b c e.2.2)
+    else upperTriangleWord L (edgeShape L a b e.1) (edgeShape L a c e.2.1) (edgeShape L b c e.2.2)
+  | _,_,_ => 1
+
+def triangleWord (s : Shape L 3) : FreeGroup (Shape L 2) :=
+  triangleWordList L s.val.1 s.property.1 s.val.2
+
+theorem shape_three_form (s : Shape L 3) :
+    ∃ (a b c : A) (h : SimplicialChains.Valid L [a,b,c]) (n : ℤ) (hn : n = -1 ∨ n = -2),
+      s = triangleShape L a b c h n hn := by
+  obtain ⟨a,b,c,hw⟩ := List.length_eq_three.mp s.property.2.1
+  have hn : s.val.2 = -1 ∨ s.val.2 = -2 := by
+    have h₁ := s.property.2.2.1
+    have h₂ := s.property.2.2.2
+    norm_num at h₂
+    omega
+  refine ⟨a,b,c,hw ▸ s.property.1,s.val.2,hn,?_⟩
+  exact Subtype.ext (Prod.ext hw rfl)
+
+include comm in
+theorem triangleWord_trivial (s : Shape L 3) :
+    wordValue (generator L φ v root hv) (triangleWord L s) = 1 := by
+  obtain ⟨a,b,c,h,n,hn,rfl⟩ := shape_three_form L s
+  rcases hn with rfl | rfl
+  · simpa [triangleWord,triangleWordList,triangleShape] using
+      (triangle_words_trivial L φ v root hv comm a b c h).1
+  · simpa [triangleWord,triangleWordList,triangleShape] using
+      (triangle_words_trivial L φ v root hv comm a b c h).2
+
+include comm in
+theorem triangleWord_chain (s : Shape L 3) (g : φ.ker) :
+    pathChain (generator L φ v root hv) (triangleWord L s) g =
+      triangleBoundary L φ v root hv (Finsupp.single (g,s) 1) := by
+  obtain ⟨a,b,c,h,n,hn,rfl⟩ := shape_three_form L s
+  rcases hn with rfl | rfl
+  · simpa [triangleWord,triangleWordList,triangleShape] using
+      (triangle_words_chain L φ v root hv comm g a b c h).1
+  · simpa [triangleWord,triangleWordList,triangleShape] using
+      (triangle_words_chain L φ v root hv comm g a b c h).2
+
+end EilenbergGanea.LevelBasis
+namespace EilenbergGanea.LevelBasis
+open CubicalChains CubicalChains.HeightFiltration
+variable {G A : Type} [Group G] [LinearOrder A]
+variable (L : SimpleGraph A) (φ : G →* Multiplicative ℤ) (v : A → G) (root : A)
+variable (hv : ∀ a, φ (v a) = Multiplicative.ofAdd (1 : ℤ))
+variable (comm : ∀ a b, L.Adj a b → Commute (v a) (v b))
+
+include comm in
+theorem triangleBoundary_eq_relatorChainMap :
+    triangleBoundary L φ v root hv =
+      relatorChainMap (generator L φ v root hv) (triangleWord L) := by
+  apply Finsupp.lhom_ext'
+  intro p
+  apply LinearMap.ext
+  intro n
+  simp only [LinearMap.coe_comp,Function.comp_apply,Finsupp.lsingle_apply,
+    relatorChainMap,Finsupp.linearCombination_single]
+  have he : Finsupp.single p n = n • Finsupp.single p (1 : ℤ) := by simp
+  rw [he,map_smul,← triangleWord_chain L φ v root hv comm p.2 p.1]
+
+variable (K : SimplicialChains.Contraction L)
+variable (exactTotal : ∀ {k : ℕ}, 0 < k → ∀ c, c ∈ total (G := G) L k → boundary v c = 0 →
+      ∃ b ∈ total (G := G) L (k+1), boundary v b = c)
+variable (dim : ∀ w, SimplicialChains.Valid L w → w.length ≤ 3)
+
+noncomputable def triangleBasis : (φ.ker × Shape L 3 →₀ ℤ) ≃ₗ[ℤ]
+    cycles (generator L φ v root hv) :=
+  LinearEquiv.ofBijective
+    ((triangleBoundary L φ v root hv).codRestrict (cycles (generator L φ v root hv))
+      (triangleBoundary_cycle L φ v root hv comm))
+    ⟨fun c d h => triangleBoundary_injective L φ v root hv comm K exactTotal dim
+      (congrArg Subtype.val h),fun c => by
+        have hc : c.val ∈ LinearMap.range (triangleBoundary L φ v root hv) := by
+          rw [triangleBoundary_range L φ v root hv comm K exactTotal]
+          exact c.property
+        obtain ⟨b,hb⟩ := hc
+        exact ⟨b,Subtype.ext hb⟩⟩
+
+theorem triangleBasis_value (c : φ.ker × Shape L 3 →₀ ℤ) :
+    (triangleBasis L φ v root hv comm K exactTotal dim c : φ.ker × Shape L 2 →₀ ℤ) =
+      relatorChainMap (generator L φ v root hv) (triangleWord L) c := by
+  change triangleBoundary L φ v root hv c = _
+  rw [triangleBoundary_eq_relatorChainMap L φ v root hv comm]
+
+end EilenbergGanea.LevelBasis
+namespace EilenbergGanea.LevelBasis
+open LevelTransport
+variable {V : Type} [LinearOrder V] (L : SimpleGraph V) (root : V) (N : ℕ)
+
+noncomputable def levelEquiv : (height L).ker ≃ NegativeLevel L N where
+  toFun := fun g => ⟨(artinGenerator L root)⁻¹ ^ N * g.val,by
+    simp only [map_mul,map_pow,map_inv,height_generator]
+    rw [show height L g.val = 1 from g.property,mul_one]
+    change N • (-1 : ℤ) = -(N : ℤ)
+    simp⟩
+  invFun := fun g => ⟨artinGenerator L root ^ N * g.val,by
+    change (height L (artinGenerator L root ^ N * g.val)).toAdd = 0
+    simp only [map_mul,map_pow,height_generator]
+    change N • (1 : ℤ) + (height L g.val).toAdd = 0
+    rw [g.property]
+    simp⟩
+  left_inv := fun g => Subtype.ext (by dsimp; group)
+  right_inv := fun g => Subtype.ext (by dsimp; group)
+
+def basedLabel (a : NegativeLevel L N → V → V → SU2)
+    (g : (height L).ker) (s : Shape L 2) : SU2 :=
+  a (levelEquiv L root N g) (s.val.1.tail.headD root) (s.val.1.headD root)
+
+@[simp] theorem basedLabel_edge (a : NegativeLevel L N → V → V → SU2)
+    (g : (height L).ker) (q r : V) (h : SimplicialChains.Valid L [q,r]) :
+    basedLabel L root N a g (edgeShape L q r h) = a (levelEquiv L root N g) r q := rfl
+
+theorem levelEquiv_edge (g : (height L).ker) (q r : V)
+    (h : SimplicialChains.Valid L [q,r]) :
+    levelEquiv L root N
+        (g * generator L (height L) (artinGenerator L) root (height_generator L) (edgeShape L q r h)) =
+      edgeVertex L N (levelEquiv L root N g) r q := by
+  have ha : L.Adj q r := by simpa [SimplicialChains.Valid] using h.2
+  apply Subtype.ext
+  simp only [levelEquiv,Equiv.coe_fn_mk,Subgroup.coe_mul,generator_edgeShape,edgeVertex]
+  rw [mul_assoc, (adjacent_generators_commute L ha).inv_left.eq]
+  group
+
+theorem shape_two_presentation (s : Shape L 2) :
+    ∃ (q r : V) (h : SimplicialChains.Valid L [q,r]), s = edgeShape L q r h := by
+  obtain ⟨q,r,hw⟩ := List.length_eq_two.mp s.property.2.1
+  have hn : s.val.2 = -1 := by
+    have h₁ := s.property.2.2.1
+    have h₂ := s.property.2.2.2
+    norm_num at h₂
+    omega
+  refine ⟨q,r,hw ▸ s.property.1,?_⟩
+  exact Subtype.ext (Prod.ext hw hn)
+
+theorem levelEquiv_edge_inv (g : (height L).ker) (q r : V)
+    (h : SimplicialChains.Valid L [q,r]) :
+    levelEquiv L root N
+        (g * (generator L (height L) (artinGenerator L) root (height_generator L) (edgeShape L q r h))⁻¹) =
+      edgeVertex L N (levelEquiv L root N g) q r := by
+  have he := levelEquiv_edge L root N
+    (g * (generator L (height L) (artinGenerator L) root (height_generator L) (edgeShape L q r h))⁻¹)
+    q r h
+  simp only [inv_mul_cancel_right] at he
+  simpa only [edgeVertex_reverse] using congrArg (fun x => edgeVertex L N x q r) he.symm
+
+variable (a : NegativeLevel L N → V → V → SU2)
+variable (rev : ∀ g q r, a (edgeVertex L N g q r) r q = (a g q r)⁻¹)
+include rev in
+theorem basedLabel_inv_edge (g : (height L).ker) (q r : V)
+    (h : SimplicialChains.Valid L [q,r]) :
+    (basedLabel L root N a
+      (g * (generator L (height L) (artinGenerator L) root (height_generator L) (edgeShape L q r h))⁻¹)
+      (edgeShape L q r h))⁻¹ = a (levelEquiv L root N g) q r := by
+  rw [basedLabel_edge,levelEquiv_edge_inv,rev,inv_inv]
+
+def orientedWord (q r : V) (h : L.Adj q r) : FreeGroup (Shape L 2) :=
+  if hlt : q < r then
+    (FreeGroup.of (edgeShape L q r (by simpa [SimplicialChains.Valid] using And.intro hlt h)))⁻¹
+  else
+    FreeGroup.of (edgeShape L r q (by
+      have hlt' : r < q := lt_of_le_of_ne (le_of_not_gt hlt) (Ne.symm (L.ne_of_adj h))
+      simpa [SimplicialChains.Valid] using And.intro hlt' h.symm))
+
+theorem orientedWord_value (q r : V) (h : L.Adj q r) :
+    (wordValue (generator L (height L) (artinGenerator L) root (height_generator L))
+      (orientedWord L q r h) : ArtinGroup L) =
+        artinGenerator L q * (artinGenerator L r)⁻¹ := by
+  unfold orientedWord
+  split_ifs
+  · simp only [map_inv,wordValue,FreeGroup.lift_apply_of,Subgroup.coe_inv,generator_edgeShape,
+      mul_inv_rev,inv_inv]
+    exact (adjacent_generators_commute L h).inv_right.symm.eq
+  · simp only [wordValue,FreeGroup.lift_apply_of,generator_edgeShape]
+    exact (adjacent_generators_commute L h).inv_right.symm.eq
+
+theorem levelEquiv_orientedWord (g : (height L).ker) (q r : V) (h : L.Adj q r) :
+    levelEquiv L root N
+      (g * wordValue (generator L (height L) (artinGenerator L) root (height_generator L))
+        (orientedWord L q r h)) = edgeVertex L N (levelEquiv L root N g) q r := by
+  apply Subtype.ext
+  simp only [levelEquiv,Equiv.coe_fn_mk,Subgroup.coe_mul,orientedWord_value,edgeVertex,mul_assoc]
+
+include rev in
+theorem orientedWord_product (g : (height L).ker) (q r : V) (h : L.Adj q r) :
+    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
+      (basedLabel L root N a) (orientedWord L q r h) g = a (levelEquiv L root N g) q r := by
+  unfold orientedWord
+  split_ifs
+  · simp only [pathProduct_inv,pathProduct_of,wordValue,FreeGroup.lift_apply_of]
+    exact basedLabel_inv_edge L root N a rev g q r _
+  · exact pathProduct_of _ _ _ _
+
+
+include rev in
+theorem inverse_edgeWord_product (g : (height L).ker) (q r : V)
+    (h : SimplicialChains.Valid L [q,r]) :
+    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
+      (basedLabel L root N a) (FreeGroup.of (edgeShape L q r h))⁻¹ g =
+      a (levelEquiv L root N g) q r := by
+  simp only [pathProduct_inv,pathProduct_of,wordValue,FreeGroup.lift_apply_of]
+  exact basedLabel_inv_edge L root N a rev g q r h
+
+variable (tri : ∀ g₀ g₁ g₂ q₀ r₀ q₁ r₁ q₂ r₂,
+    edgeVertex L N g₀ q₀ r₀ = g₁ → edgeVertex L N g₁ q₁ r₁ = g₂ →
+      edgeVertex L N g₂ q₂ r₂ = g₀ → a g₀ q₀ r₀ * a g₁ q₁ r₁ * a g₂ q₂ r₂ = 1)
+
+include rev tri in
+theorem lowerTriangleWord_product (g : (height L).ker) (q r s : V)
+    (h : SimplicialChains.Valid L [q,r,s]) :
+    let e := valid_triangle_edges L h
+    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
+      (basedLabel L root N a)
+      (lowerTriangleWord L (edgeShape L q r e.1) (edgeShape L q s e.2.1)
+        (edgeShape L r s e.2.2)) g = 1 := by
+  let gen := generator L (height L) (artinGenerator L) root (height_generator L)
+  let e := valid_triangle_edges L h
+  let ab := edgeShape L q r e.1
+  let ac := edgeShape L q s e.2.1
+  let bc := edgeShape L r s e.2.2
+  have he : gen ab * gen bc = gen ac :=
+    Subtype.ext (edge_values_compose L (artinGenerator L)
+      (fun _ _ hh => adjacent_generators_commute L hh) q r s h).1
+  have hc : gen ac * (gen bc)⁻¹ = gen ab := by rw [← he]; group
+  have hc' : g * gen ac * (gen bc)⁻¹ = g * gen ab := by rw [mul_assoc,hc]
+  have ha : g * gen ab * (gen ab)⁻¹ = g := by group
+  have T := tri (levelEquiv L root N g) (levelEquiv L root N (g * gen ac))
+    (levelEquiv L root N (g * gen ab)) s q r s q r
+    (levelEquiv_edge L root N g q s e.2.1).symm
+    (by rw [← levelEquiv_edge_inv L root N (g * gen ac) r s e.2.2]; exact congrArg _ hc')
+    (by rw [← levelEquiv_edge_inv L root N (g * gen ab) q r e.1]; exact congrArg _ ha)
+  change pathProduct gen (basedLabel L root N a) (lowerTriangleWord L ab ac bc) g = 1
+  simp only [lowerTriangleWord,pathProduct_mul,pathProduct_of,map_mul,map_inv,
+    wordValue,FreeGroup.lift_apply_of]
+  rw [hc]
+  rw [inverse_edgeWord_product L root N a rev _ r s e.2.2,
+    inverse_edgeWord_product L root N a rev _ q r e.1]
+  exact T
+
+include rev tri in
+theorem upperInnerWord_product (g : (height L).ker) (q r s : V)
+    (h : SimplicialChains.Valid L [q,r,s]) :
+    let e := valid_triangle_edges L h
+    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
+      (basedLabel L root N a)
+      (FreeGroup.of (edgeShape L r s e.2.2) * FreeGroup.of (edgeShape L q r e.1) *
+        (FreeGroup.of (edgeShape L q s e.2.1))⁻¹) g = 1 := by
+  let gen := generator L (height L) (artinGenerator L) root (height_generator L)
+  let e := valid_triangle_edges L h
+  let ab := edgeShape L q r e.1
+  let ac := edgeShape L q s e.2.1
+  let bc := edgeShape L r s e.2.2
+  have he : gen bc * gen ab = gen ac :=
+    Subtype.ext (edge_values_compose L (artinGenerator L)
+      (fun _ _ hh => adjacent_generators_commute L hh) q r s h).2
+  have hc : g * gen bc * gen ab = g * gen ac := by rw [mul_assoc,he]
+  have ha : g * gen ac * (gen ac)⁻¹ = g := by group
+  have T := tri (levelEquiv L root N g) (levelEquiv L root N (g * gen bc))
+    (levelEquiv L root N (g * gen ac)) s r r q q s
+    (levelEquiv_edge L root N g r s e.2.2).symm
+    (by rw [← levelEquiv_edge L root N (g * gen bc) q r e.1]; exact congrArg _ hc)
+    (by rw [← levelEquiv_edge_inv L root N (g * gen ac) q s e.2.1]; exact congrArg _ ha)
+  change pathProduct gen (basedLabel L root N a)
+    (FreeGroup.of bc * FreeGroup.of ab * (FreeGroup.of ac)⁻¹) g = 1
+  simp only [pathProduct_mul,pathProduct_of,map_mul,wordValue,FreeGroup.lift_apply_of]
+  rw [he,inverse_edgeWord_product L root N a rev _ q s e.2.1]
+  exact T
+
+include rev tri in
+theorem upperTriangleWord_product (g : (height L).ker) (q r s : V)
+    (h : SimplicialChains.Valid L [q,r,s]) :
+    let e := valid_triangle_edges L h
+    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
+      (basedLabel L root N a)
+      (upperTriangleWord L (edgeShape L q r e.1) (edgeShape L q s e.2.1)
+        (edgeShape L r s e.2.2)) g = 1 := by
+  let gen := generator L (height L) (artinGenerator L) root (height_generator L)
+  let e := valid_triangle_edges L h
+  let ab := edgeShape L q r e.1
+  let ac := edgeShape L q s e.2.1
+  let bc := edgeShape L r s e.2.2
+  have he : gen bc * gen ab = gen ac :=
+    Subtype.ext (edge_values_compose L (artinGenerator L)
+      (fun _ _ hh => adjacent_generators_commute L hh) q r s h).2
+  have hr : wordValue gen (FreeGroup.of bc * FreeGroup.of ab * (FreeGroup.of ac)⁻¹) = 1 := by
+    simp only [map_mul,map_inv,wordValue,FreeGroup.lift_apply_of,he,mul_inv_cancel]
+  change pathProduct gen (basedLabel L root N a) (upperTriangleWord L ab ac bc) g = 1
+  rw [upperTriangleWord,pathProduct_conjugate_loop gen _ _ _ hr]
+  rw [upperInnerWord_product L root N a rev tri _ q r s h]
+  group
+
+include rev tri in
+theorem triangleWord_product (s : Shape L 3) (g : (height L).ker) :
+    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
+      (basedLabel L root N a) (triangleWord L s) g = 1 := by
+  obtain ⟨q,r,t,h,n,hn,rfl⟩ := shape_three_form L s
+  rcases hn with rfl | rfl
+  · simpa [triangleWord,triangleWordList,triangleShape] using
+      lowerTriangleWord_product L root N a rev tri g q r t h
+  · simpa [triangleWord,triangleWordList,triangleShape] using
+      upperTriangleWord_product L root N a rev tri g q r t h
+
+
+variable {L N}
+def walkWord {g k : NegativeLevel L N} : Walk L N g k → FreeGroup (Shape L 2)
+  | .nil _ => 1
+  | .cons q r h p => orientedWord L q r h * walkWord p
+
+ theorem walkWord_endpoint {g k : NegativeLevel L N} (p : Walk L N g k)
+    (x : (height L).ker) (hx : levelEquiv L root N x = g) :
+    levelEquiv L root N
+      (x * wordValue (generator L (height L) (artinGenerator L) root (height_generator L))
+        (walkWord p)) = k := by
+  induction p generalizing x with
+  | nil g => simpa [walkWord] using hx
+  | @cons g k q r h p ih =>
+    simp only [walkWord,map_mul,← mul_assoc]
+    exact ih _ ((levelEquiv_orientedWord L root N x q r h).trans
+      (congrArg (fun t => edgeVertex L N t q r) hx))
+
+include rev in
+ theorem walkWord_product {g k : NegativeLevel L N} (p : Walk L N g k)
+    (x : (height L).ker) (hx : levelEquiv L root N x = g) :
+    pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
+      (basedLabel L root N a) (walkWord p) x = p.eval a := by
+  induction p generalizing x with
+  | nil g => simp [walkWord,Walk.eval]
+  | @cons g k q r h p ih =>
+    rw [walkWord,pathProduct_mul,orientedWord_product L root N a rev x q r h]
+    rw [ih _ ((levelEquiv_orientedWord L root N x q r h).trans
+      (congrArg (fun t => edgeVertex L N t q r) hx)),hx]
+    rfl
+
+include rev in
+ theorem nontrivial_word_of_walk {g : NegativeLevel L N} (p : Walk L N g g)
+    (hp : p.eval a ≠ 1) :
+    ∃ (w : FreeGroup (Shape L 2)) (x : (height L).ker),
+      wordValue (generator L (height L) (artinGenerator L) root (height_generator L)) w = 1 ∧
+      pathProduct (generator L (height L) (artinGenerator L) root (height_generator L))
+        (basedLabel L root N a) w x ≠ 1 := by
+  let x := (levelEquiv L root N).symm g
+  have hx : levelEquiv L root N x = g := (levelEquiv L root N).apply_symm_apply g
+  refine ⟨walkWord p,x,?_,?_⟩
+  · have he := walkWord_endpoint root p x hx
+    have he' := (levelEquiv L root N).injective (he.trans hx.symm)
+    exact mul_left_cancel (he'.trans (mul_one x).symm)
+  · rwa [walkWord_product root a rev p x hx]
+
+end EilenbergGanea.LevelBasis
+
+namespace EilenbergGanea.LevelBasis
+open LevelTransport
+variable {V : Type} [LinearOrder V] [Fintype V] [Nonempty V] (L : SimpleGraph V)
+variable [Group.ResiduallyFinite (height L).ker]
+variable (root : V) (hconn : L.Preconnected)
+variable (K : SimplicialChains.Contraction L)
+variable (dim : ∀ w, SimplicialChains.Valid L w → w.length ≤ 3)
+
+include root hconn K dim in
+/-- The source obstruction's algebraic endpoint, with no cardinal bound on
+an alleged presentation. Its geometric premise must still be derived from an
+ordinary two-dimensional classifying CW space. -/
+theorem no_boundary_basis_presentation
+    (small : ∀ ε : ℝ, 0 < ε →
+      ∃ N : ℕ, 1 < N ∧ ∃ a : NegativeLevel L N → V → V → SU2,
+      (∀ g q r, a (edgeVertex L N g q r) r q = (a g q r)⁻¹) ∧
+      (∀ g q r, dist (a g q r) 1 < ε) ∧
+      (∀ g₀ g₁ g₂ q₀ r₀ q₁ r₁ q₂ r₂,
+        edgeVertex L N g₀ q₀ r₀ = g₁ →
+        edgeVertex L N g₁ q₁ r₁ = g₂ →
+        edgeVertex L N g₂ q₂ r₂ = g₀ →
+        a g₀ q₀ r₀ * a g₁ q₁ r₁ * a g₂ q₂ r₂ = 1) ∧
+      ∃ (g : NegativeLevel L N) (p : Walk L N g g), p.eval a ≠ 1)
+    {A R : Type*} (a₀ : A → (height L).ker)
+    (ha₀ : Function.Surjective (wordValue a₀))
+    (r₀ : R → FreeGroup A) (hr₀ : ∀ i, wordValue a₀ (r₀ i) = 1)
+    (hp₀ : (wordValue a₀).ker = Subgroup.normalClosure (Set.range r₀))
+    (E : BasedChains (height L).ker R ≃ₗ[ℤ] cycles a₀)
+    (hE : ∀ c, (E c : BasedChains (height L).ker A) = relatorChainMap a₀ r₀ c) : False := by
+  let gen := generator L (height L) (artinGenerator L) root (height_generator L)
+  let comm := fun q r h => @adjacent_generators_commute V L q r h
+  let B := triangleBasis L (height L) (artinGenerator L) root (height_generator L)
+    comm K (artin_total_cycle_bounds L) dim
+  obtain ⟨η,hη,hηa⟩ := boundary_basis_small_transport gen
+    (generator_surjective L hconn root) (triangleWord L)
+    (triangleWord_trivial L (height L) (artinGenerator L) root (height_generator L) comm)
+    B (triangleBasis_value L (height L) (artinGenerator L) root (height_generator L)
+      comm K (artin_total_cycle_bounds L) dim) a₀ ha₀ r₀ hr₀ hp₀ E hE
+  obtain ⟨N,_,a,ha,hb,ht,g,p,hp⟩ := small η hη
+  obtain ⟨w,x,hw,hx⟩ := nontrivial_word_of_walk root a ha p hp
+  apply hx
+  exact hηa (basedLabel L root N a)
+    (fun x s => by
+      simpa only [basedLabel,Subtype.dist_eq,dist_eq_norm,OneMemClass.coe_one] using
+        hb (levelEquiv L root N x) (s.val.1.tail.headD root) (s.val.1.headD root))
+    (triangleWord_product L root N a ha ht) w hw x
+
+end EilenbergGanea.LevelBasis
+
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/LevelTransport.lean

@@ -1,626 +1,626 @@
-import OAI.Topology.EilenbergGanea.AffineTransport
-import OAI.Topology.EilenbergGanea.SeedConnection
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set Filter Topology MeasureTheory
-open scoped Quaternion ContDiff
-
-namespace EilenbergGanea
-namespace LevelTransport
-open TraceWords AffineTransport
-
-variable {V : Type*} [Fintype V] [LinearOrder V] (L : SimpleGraph V)
-variable (hdim : ∀ s : Finset V, (s : Set V).Pairwise L.Adj → s.card ≤ 3)
-
-def point (g : ArtinGroup L) (hg : 0 < traceLength L (artinNormalTrace L g)) : TrianglePoint V :=
-  ⟨artinDirection L g,by
-    apply mem_triangleSet.mpr
-    refine ⟨(traceCounts L (artinNormalTrace L g)).support,
-      hdim _ (traceCounts_support_clique L _),⟨?_,?_⟩,?_⟩
-    · exact traceDirection_nonneg L _
-    · exact traceDirection_sum L hg
-    · intro v hv
-      simp only [Finsupp.notMem_support_iff] at hv
-      simp [artinDirection,traceDirection,hv]⟩
-
-@[simp] theorem point_val (g : ArtinGroup L) (hg : 0 < traceLength L (artinNormalTrace L g)) :
-    (point L hdim g hg : V → ℝ) = artinDirection L g := rfl
-
-theorem point_mem_support (g : ArtinGroup L) (hg : 0 < traceLength L (artinNormalTrace L g))
-    {v : V} (hv : (point L hdim g hg : V → ℝ) v ≠ 0) :
-    v ∈ (traceCounts L (artinNormalTrace L g)).support := by
-  by_contra h
-  have he := Finsupp.notMem_support_iff.mp h
-  exact hv (by simp [point,artinDirection,traceDirection,he])
-
-theorem point_clique (g : ArtinGroup L) (hg : 0 < traceLength L (artinNormalTrace L g)) :
-    CliquePoint L (point L hdim g hg) := by
-  intro a ha v hv
-  by_cases hav : a = v
-  · exact Or.inl hav
-  · exact Or.inr (traceCounts_support_clique L _
-      (point_mem_support L hdim g hg ha) (point_mem_support L hdim g hg hv) hav)
-
-/-- Both hypotheses refer to actual reduced Artin lengths, not an assumed word metric. -/
-theorem point_step (g : ArtinGroup L) (x : Letter V)
-    (hg : 0 < traceLength L (artinNormalTrace L g))
-    (hgx : 0 < traceLength L (artinNormalTrace L (g * letterValue (artinGenerator L) x)))
-    (N : ℕ) (hNg : N ≤ traceLength L (artinNormalTrace L g))
-    (hNgx : N ≤ traceLength L (artinNormalTrace L (g * letterValue (artinGenerator L) x))) :
-    Compatible L (point L hdim g hg) (point L hdim (g * letterValue (artinGenerator L) x) hgx) ∧
-    dist (point L hdim g hg) (point L hdim (g * letterValue (artinGenerator L) x) hgx) ≤
-      2 / (N+1 : ℝ) := by
-  classical
-  obtain ⟨hs,n,_,hn,hbd⟩ := artinDirection_right_step L g x hg hgx
-  have hNn : N ≤ n := by rcases hn with ⟨he,_⟩ | ⟨he,_⟩ <;> omega
-  constructor
-  · intro v w hv hw
-    by_cases he : v = w
-    · exact Or.inl he
-    · apply Or.inr
-      apply hs _ _ he
-      · rcases hv with hv | hv
-        · exact Finset.mem_union_left _ (point_mem_support L hdim g hg hv)
-        · exact Finset.mem_union_right _ (point_mem_support L hdim _ hgx hv)
-      · rcases hw with hw | hw
-        · exact Finset.mem_union_left _ (point_mem_support L hdim g hg hw)
-        · exact Finset.mem_union_right _ (point_mem_support L hdim _ hgx hw)
-  · have hc : (N : ℝ) + 1 ≤ (n : ℝ) + 1 := by exact_mod_cast Nat.add_le_add_right hNn 1
-    have hd : 0 < (N : ℝ) + 1 := by positivity
-    have hfrac : 2 / (n+1 : ℝ) ≤ 2 / (N+1 : ℝ) :=
-      div_le_div_of_nonneg_left (by norm_num) hd hc
-    apply le_trans ?_ hfrac
-    change dist (artinDirection L g) (artinDirection L (g * letterValue (artinGenerator L) x)) ≤ _
-    apply (dist_pi_le_iff (by positivity)).mpr
-    intro v
-    rw [Real.dist_eq,abs_sub_comm]
-    exact le_trans (Finset.single_le_sum (fun w _ => abs_nonneg (artinDirection L (g * letterValue (artinGenerator L) x) w - artinDirection L g w)) (Finset.mem_univ v)) hbd
-
-
-/-- The integral level downstairs; the action is by left multiplication by the height kernel. -/
-abbrev NegativeLevel (N : ℕ) := {g : ArtinGroup L // (height L g).toAdd = -(N : ℤ)}
-
-omit [Fintype V] [LinearOrder V] in
-theorem negative_level_length (N : ℕ) {g : ArtinGroup L}
-    (hg : (height L g).toAdd = -(N : ℤ)) :
-    N ≤ traceLength L (artinNormalTrace L g) := by
-  have hh := height_abs_le_traceLength L g
-  rw [hg,abs_neg,abs_of_nonneg (Nat.cast_nonneg N)] at hh
-  exact_mod_cast hh
-
-def levelPoint (N : ℕ) (hN : 0 < N) (g : NegativeLevel L N) : TrianglePoint V :=
-  point L hdim g (lt_of_lt_of_le hN (negative_level_length L N g.property))
-
-@[simp] theorem levelPoint_val (N : ℕ) (hN : 0 < N) (g : NegativeLevel L N) :
-    (levelPoint L hdim N hN g : V → ℝ) = artinDirection L g := rfl
-
-def upperVertex (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) (v : V) :
-    NegativeLevel L (N-1) :=
-  ⟨(g : ArtinGroup L) * artinGenerator L v,by
-    rw [map_mul,height_generator]
-    change (height L (g : ArtinGroup L)).toAdd + 1 = _
-    rw [g.property]
-    omega⟩
-
-def edgeVertex (N : ℕ) (g : NegativeLevel L N) (q r : V) : NegativeLevel L N :=
-  ⟨(g : ArtinGroup L) * artinGenerator L q * (artinGenerator L r)⁻¹,by
-    simp only [map_mul,map_inv,height_generator]
-    change (height L (g : ArtinGroup L)).toAdd + 1 + -1 = _
-    rw [g.property]
-    omega⟩
-
-omit [Fintype V] [LinearOrder V] in
-theorem edgeVertex_reverse (N : ℕ) (g : NegativeLevel L N) (q r : V) :
-    edgeVertex L N (edgeVertex L N g q r) r q = g := by
-  apply Subtype.ext
-  change (g : ArtinGroup L) * _ * _ * _ * _ = _
-  group
-
-omit [Fintype V] [LinearOrder V] in
-theorem upperVertex_reverse (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) (q r : V) :
-    upperVertex L N hN (edgeVertex L N g q r) r = upperVertex L N hN g q := by
-  apply Subtype.ext
-  dsimp only [edgeVertex,upperVertex]
-  group
-
-theorem upperPoint_step (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) (v : V) :
-    Compatible L (levelPoint L hdim N (by omega) g)
-      (levelPoint L hdim (N-1) (by omega) (upperVertex L N hN g v)) ∧
-    dist (levelPoint L hdim N (by omega) g)
-      (levelPoint L hdim (N-1) (by omega) (upperVertex L N hN g v)) ≤ 2 / (N : ℝ) := by
-  let hu := upperVertex L N hN g v
-  have hlen := negative_level_length L N g.property
-  have hlen' := negative_level_length L (N-1) hu.property
-  dsimp only [hu,upperVertex] at hlen'
-  have H := point_step L hdim (g : ArtinGroup L) (v,true)
-    (lt_of_lt_of_le (by omega : 0 < N) hlen)
-    (by simpa only [letterValue_positive] using lt_of_lt_of_le (by omega : 0 < N-1) hlen')
-    (N-1) (le_trans (Nat.sub_le N 1) hlen)
-    (by simpa only [letterValue_positive] using hlen')
-  have he : ((N-1 : ℕ) : ℝ) + 1 = (N : ℝ) := by exact_mod_cast (show (N-1 : ℕ) + 1 = N by omega)
-  simpa only [levelPoint,upperVertex,letterValue_positive,he] using H
-
-variable (h : V → V → SU2) (Q : V → ℍ)
-
-def edgeLabel (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) (q r : V) : SU2 :=
-  edgeTransport h Q (levelPoint L hdim N (by omega) g)
-    (levelPoint L hdim (N-1) (by omega) (upperVertex L N hN g q))
-    (levelPoint L hdim N (by omega) (edgeVertex L N g q r))
-
-theorem edgeLabel_reverse (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
-    (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) (q r : V) :
-    edgeLabel L hdim h Q N hN (edgeVertex L N g q r) r q =
-      (edgeLabel L hdim h Q N hN g q r)⁻¹ := by
-  have h₁ := (upperPoint_step L hdim N hN g q).1
-  have h₂ := (upperPoint_step L hdim N hN (edgeVertex L N g q r) r).1
-  rw [upperVertex_reverse] at h₂
-  unfold edgeLabel
-  rw [edgeVertex_reverse,upperVertex_reverse]
-  exact edgeTransport_reverse L h Q hf hQ h₁ (compatible_symm L h₂)
-
-theorem edgeLabel_uniform (hQ : GeneralPosition h Q) {ε : ℝ} (hε : 0 < ε) :
-    ∃ M : ℕ, ∀ N : ℕ, M ≤ N → ∀ hN : 1 < N, ∀ g : NegativeLevel L N, ∀ q r : V,
-      dist (edgeLabel L hdim h Q N hN g q r) 1 < ε := by
-  obtain ⟨δ,hδ,H⟩ := edgeTransport_uniform h Q hQ hε
-  obtain ⟨M,hM⟩ := exists_nat_gt (2 / δ)
-  refine ⟨M,?_⟩
-  intro N hMN hN g q r
-  have hN0 : (0 : ℝ) < N := by exact_mod_cast (by omega : 0 < N)
-  have hbound : 2 / (N : ℝ) < δ := by
-    apply (div_lt_iff₀ hN0).mpr
-    have hMN' : (M : ℝ) ≤ N := by exact_mod_cast hMN
-    have hh := (div_lt_iff₀ hδ).mp hM
-    nlinarith
-  apply H
-  · exact lt_of_le_of_lt (upperPoint_step L hdim N hN g q).2 hbound
-  · have hh := (upperPoint_step L hdim N hN (edgeVertex L N g q r) r).2
-    rw [upperVertex_reverse,dist_comm] at hh
-    exact lt_of_le_of_lt hh hbound
-
-
-omit [LinearOrder V] in
-/-- Six compatible generator pieces around a level triangle lie in one flat chart
-once the three edge images are uniformly short. -/
-theorem triangle_of_short (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
-    (b c d u v w : TrianglePoint V) (δ : ℝ) (hδ : 0 < δ)
-    (hsmall : 4 * δ < 1 / (Fintype.card V + 1 : ℝ))
-    (hb : CliquePoint L b) (hc : CliquePoint L c) (hd : CliquePoint L d)
-    (hu : CliquePoint L u) (hv : CliquePoint L v) (hw : CliquePoint L w)
-    (hbu : Compatible L b u) (huc : Compatible L u c)
-    (hcv : Compatible L c v) (hvd : Compatible L v d)
-    (hdw : Compatible L d w) (hwb : Compatible L w b)
-    (dbu : dist b u ≤ δ) (duc : dist u c ≤ δ)
-    (dcv : dist c v ≤ δ) (dvd : dist v d ≤ δ)
-    (_ddw : dist d w ≤ δ) (dwb : dist w b ≤ δ) :
-    edgeTransport h Q b u c * edgeTransport h Q c v d * edgeTransport h Q d w b = 1 := by
-  have dbc : dist b c ≤ 2 * δ := by linarith [dist_triangle b u c]
-  have dbv : dist b v ≤ 3 * δ := by linarith [dist_triangle b c v]
-  have dbd : dist b d ≤ 4 * δ := by linarith [dist_triangle b v d]
-  have dbw : dist b w ≤ δ := by simpa only [dist_comm] using dwb
-  obtain ⟨a,ha⟩ := exists_large_coordinate b
-  apply edgeTransport_triangle L h Q hf hQ b c d u v w hbu huc hcv hvd hdw hwb
-  · exact nearby_star L ha hb (by simpa using lt_of_lt_of_le hδ (le_of_lt (by linarith : δ < 1 / (Fintype.card V + 1 : ℝ))))
-  · exact nearby_star L ha hc (by linarith)
-  · exact nearby_star L ha hd (lt_of_le_of_lt dbd hsmall)
-  · exact nearby_star L ha hu (by linarith)
-  · exact nearby_star L ha hv (by linarith)
-  · exact nearby_star L ha hw (by linarith)
-
-/-- Every oriented three-edge loop with these actual level endpoints is flat.
-In particular both kinds of sections of a three-cube are covered. -/
-theorem edgeLabel_triangle (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
-    (N : ℕ) (hN : 1 < N) (hsize : 8 / (N : ℝ) < 1 / (Fintype.card V + 1 : ℝ))
-    (g₀ g₁ g₂ : NegativeLevel L N) (q₀ r₀ q₁ r₁ q₂ r₂ : V)
-    (e₀ : edgeVertex L N g₀ q₀ r₀ = g₁)
-    (e₁ : edgeVertex L N g₁ q₁ r₁ = g₂)
-    (e₂ : edgeVertex L N g₂ q₂ r₂ = g₀) :
-    edgeLabel L hdim h Q N hN g₀ q₀ r₀ *
-      edgeLabel L hdim h Q N hN g₁ q₁ r₁ *
-      edgeLabel L hdim h Q N hN g₂ q₂ r₂ = 1 := by
-  have h₀ := upperPoint_step L hdim N hN g₀ q₀
-  have h₁ := upperPoint_step L hdim N hN g₁ q₁
-  have h₂ := upperPoint_step L hdim N hN g₂ q₂
-  have j₀ := upperPoint_step L hdim N hN g₁ r₀
-  have j₁ := upperPoint_step L hdim N hN g₂ r₁
-  have j₂ := upperPoint_step L hdim N hN g₀ r₂
-  rw [← e₀,upperVertex_reverse,e₀] at j₀
-  rw [← e₁,upperVertex_reverse,e₁] at j₁
-  rw [← e₂,upperVertex_reverse,e₂] at j₂
-  unfold edgeLabel
-  rw [e₀,e₁,e₂]
-  apply triangle_of_short L h Q hf hQ _ _ _ _ _ _ (2/(N : ℝ))
-  · positivity
-  · convert hsize using 1; ring
-  · exact point_clique L hdim _ _
-  · exact point_clique L hdim _ _
-  · exact point_clique L hdim _ _
-  · exact point_clique L hdim _ _
-  · exact point_clique L hdim _ _
-  · exact point_clique L hdim _ _
-  · exact h₀.1
-  · exact compatible_symm L j₀.1
-  · exact h₁.1
-  · exact compatible_symm L j₁.1
-  · exact h₂.1
-  · exact compatible_symm L j₂.1
-  · exact h₀.2
-  · simpa only [dist_comm] using j₀.2
-  · exact h₁.2
-  · simpa only [dist_comm] using j₁.2
-  · exact h₂.2
-  · simpa only [dist_comm] using j₂.2
-
-
-/-- A vertex of the simplex is also an honest point of its two-skeleton. -/
-def vertexPoint (v : V) : TrianglePoint V :=
-  ⟨fun w => if v = w then 1 else 0,by
-    apply mem_triangleSet.mpr
-    refine ⟨{v},by simp,⟨?_,?_⟩,?_⟩
-    · intro w; dsimp only; split_ifs <;> norm_num
-    · simp
-    · intro w hw
-      simp only [Finset.mem_singleton] at hw
-      simp [Ne.symm hw]⟩
-
-@[simp] theorem vertexPoint_val (v w : V) :
-    (vertexPoint v : V → ℝ) w = if v = w then 1 else 0 := rfl
-
-theorem star_vertex {a v : V} (hav : Related L a v) : Star L a (vertexPoint v) := by
-  intro w hw
-  have he : v = w := by by_contra hh; exact hw (by simp [vertexPoint,hh])
-  simpa only [← he] using hav
-
-omit [Fintype V] [LinearOrder V] in
-theorem negative_powers_upper {q r : V} (hqr : L.Adj q r) (n m : ℕ) :
-    (artinGenerator L q)⁻¹ ^ (n+1) * (artinGenerator L r)⁻¹ ^ m * artinGenerator L q =
-      (artinGenerator L q)⁻¹ ^ n * (artinGenerator L r)⁻¹ ^ m := by
-  have hc := (adjacent_generators_commute L hqr).inv_right.pow_right m
-  calc
-    _ = (artinGenerator L q)⁻¹ ^ n *
-        ((artinGenerator L q)⁻¹ * ((artinGenerator L r)⁻¹ ^ m * artinGenerator L q)) := by
-      simp only [pow_succ]; group
-    _ = (artinGenerator L q)⁻¹ ^ n *
-        ((artinGenerator L q)⁻¹ * (artinGenerator L q * (artinGenerator L r)⁻¹ ^ m)) := by
-      rw [← hc.eq]
-    _ = _ := by group
-
-/-- Negative commuting powers remain exactly in their two-coordinate face. -/
-theorem powersPoint_star {q r : V} (hqr : L.Adj q r) (n m : ℕ)
-    (hg : 0 < traceLength L (artinNormalTrace L
-      ((artinGenerator L q)⁻¹ ^ n * (artinGenerator L r)⁻¹ ^ m))) :
-    Star L q (point L hdim ((artinGenerator L q)⁻¹ ^ n * (artinGenerator L r)⁻¹ ^ m) hg) := by
-  intro v hv
-  by_cases hqv : q = v
-  · exact Or.inl hqv
-  by_cases hrv : r = v
-  · exact Or.inr (hrv ▸ hqr)
-  exfalso
-  apply hv
-  rw [point_val,direction_negative_commuting_powers L hqr]
-  simp [hqv,hrv]
-
-/-- Constant endpoint in the q-direction, stated with a concrete positive height. -/
-theorem powersPoint_first {q r : V} (hqr : L.Adj q r) (N : ℕ) (hN : 0 < N)
-    (hg : 0 < traceLength L (artinNormalTrace L
-      ((artinGenerator L q)⁻¹ ^ N * (artinGenerator L r)⁻¹ ^ 0))) :
-    point L hdim ((artinGenerator L q)⁻¹ ^ N * (artinGenerator L r)⁻¹ ^ 0) hg =
-      vertexPoint q := by
-  apply Subtype.ext
-  funext v
-  rw [point_val,vertexPoint_val,direction_negative_commuting_powers L hqr]
-  have hn : (N : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hN
-  by_cases hv : q = v <;> simp [hv,hn]
-
-theorem powersPoint_last {q r : V} (hqr : L.Adj q r) (N : ℕ) (hN : 0 < N)
-    (hg : 0 < traceLength L (artinNormalTrace L
-      ((artinGenerator L q)⁻¹ ^ 0 * (artinGenerator L r)⁻¹ ^ N))) :
-    point L hdim ((artinGenerator L q)⁻¹ ^ 0 * (artinGenerator L r)⁻¹ ^ N) hg =
-      vertexPoint r := by
-  apply Subtype.ext
-  funext v
-  rw [point_val,vertexPoint_val,direction_negative_commuting_powers L hqr]
-  have hn : (N : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hN
-  by_cases hv : r = v <;> simp [hv,hn]
-
-def powerVertex (N : ℕ) (q r : V) (j : ℕ) : NegativeLevel L N :=
-  ⟨(artinGenerator L q)⁻¹ ^ (N - min j N) * (artinGenerator L r)⁻¹ ^ min j N,by
-    simp only [map_mul,map_pow,map_inv,height_generator]
-    change (N-min j N : ℕ) • (-1 : ℤ) + (min j N : ℕ) • (-1 : ℤ) = _
-    simp only [nsmul_eq_mul,mul_neg,mul_one]
-    have hh : min j N ≤ N := min_le_right _ _
-    omega⟩
-
-omit [Fintype V] [LinearOrder V] in
-theorem powerVertex_step (N : ℕ) {q r : V} (hqr : L.Adj q r) {j : ℕ} (hj : j < N) :
-    edgeVertex L N (powerVertex L N q r j) q r = powerVertex L N q r (j+1) := by
-  apply Subtype.ext
-  dsimp only [powerVertex,edgeVertex]
-  rw [min_eq_left (by omega : j ≤ N),min_eq_left (by omega : j+1 ≤ N)]
-  have he : N-j = (N-(j+1))+1 := by omega
-  rw [he,negative_powers_upper L hqr,pow_succ]
-  group
-
-omit [Fintype V] [LinearOrder V] in
-theorem powerVertex_upper (N : ℕ) (hN : 1 < N) {q r : V}
-    (hqr : L.Adj q r) {j : ℕ} (hj : j < N) :
-    (upperVertex L N hN (powerVertex L N q r j) q : ArtinGroup L) =
-      (artinGenerator L q)⁻¹ ^ (N-(j+1)) * (artinGenerator L r)⁻¹ ^ j := by
-  dsimp only [powerVertex,upperVertex]
-  rw [min_eq_left (by omega : j ≤ N)]
-  have he : N-j = (N-(j+1))+1 := by omega
-  rw [he,negative_powers_upper L hqr]
-
-theorem powerVertex_star (N : ℕ) (hN : 0 < N) {q r : V} (hqr : L.Adj q r) (j : ℕ) :
-    Star L q (levelPoint L hdim N hN (powerVertex L N q r j)) :=
-  powersPoint_star L hdim hqr _ _ _
-
-theorem powerVertex_upper_star (N : ℕ) (hN : 1 < N) {q r : V}
-    (hqr : L.Adj q r) {j : ℕ} (hj : j < N) :
-    Star L q (levelPoint L hdim (N-1) (by omega)
-      (upperVertex L N hN (powerVertex L N q r j) q)) := by
-  unfold levelPoint
-  generalize_proofs hpos
-  simp only [powerVertex_upper L N hN hqr hj] at hpos ⊢
-  exact powersPoint_star L hdim hqr _ _ _
-
-theorem powerLabel_chart (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
-    (N : ℕ) (hN : 1 < N) {q r : V} (hqr : L.Adj q r) {j : ℕ} (hj : j < N) :
-    edgeLabel L hdim h Q N hN (powerVertex L N q r j) q r =
-      chartTransport h Q q (levelPoint L hdim N (by omega) (powerVertex L N q r j))
-        (levelPoint L hdim N (by omega) (powerVertex L N q r (j+1))) := by
-  have hb := upperPoint_step L hdim N hN (powerVertex L N q r j) q
-  have hc := upperPoint_step L hdim N hN (edgeVertex L N (powerVertex L N q r j) q r) r
-  rw [upperVertex_reverse] at hc
-  unfold edgeLabel
-  rw [edgeTransport_chart L h Q hf hQ hb.1 (compatible_symm L hc.1)
-    (powerVertex_star L hdim N (by omega) hqr j) (powerVertex_upper_star L hdim N hN hqr hj)]
-  · rw [powerVertex_step L N hqr hj]
-  · rw [powerVertex_step L N hqr hj]
-    exact powerVertex_star L hdim N (by omega) hqr (j+1)
-
-
-/-- Directed walks retain the chosen generator pair, including parallel edges. -/
-inductive Walk (N : ℕ) : NegativeLevel L N → NegativeLevel L N → Type _
-  | nil (g : NegativeLevel L N) : Walk N g g
-  | cons {g k : NegativeLevel L N} (q r : V) (hqr : L.Adj q r)
-      (p : Walk N (edgeVertex L N g q r) k) : Walk N g k
-
-namespace Walk
-variable {L} {N : ℕ}
-
-def append {g k l : NegativeLevel L N} : Walk L N g k → Walk L N k l → Walk L N g l
-  | .nil _, p => p
-  | .cons q r hqr p, t => .cons q r hqr (append p t)
-
-def cast {g k g' k' : NegativeLevel L N} (p : Walk L N g k)
-    (hg : g = g') (hk : k = k') : Walk L N g' k' := hg ▸ hk ▸ p
-
-end Walk
-
-def walkProduct (N : ℕ) (hN : 1 < N) {g k : NegativeLevel L N} : Walk L N g k → SU2
-  | .nil _ => 1
-  | .cons (g := g) q r _ p => edgeLabel L hdim h Q N hN g q r * walkProduct N hN p
-
-@[simp] theorem walkProduct_nil (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) :
-    walkProduct L hdim h Q N hN (.nil g) = 1 := rfl
-
-@[simp] theorem walkProduct_cons (N : ℕ) (hN : 1 < N) {g k : NegativeLevel L N}
-    (q r : V) (hqr : L.Adj q r) (p : Walk L N (edgeVertex L N g q r) k) :
-    walkProduct L hdim h Q N hN (.cons q r hqr p) =
-      edgeLabel L hdim h Q N hN g q r * walkProduct L hdim h Q N hN p := rfl
-
-@[simp] theorem walkProduct_append (N : ℕ) (hN : 1 < N) {g k l : NegativeLevel L N}
-    (p : Walk L N g k) (t : Walk L N k l) :
-    walkProduct L hdim h Q N hN (p.append t) =
-      walkProduct L hdim h Q N hN p * walkProduct L hdim h Q N hN t := by
-  induction p with
-  | nil => simp [Walk.append]
-  | cons q r hqr p ih => simp [Walk.append,ih,mul_assoc]
-
-@[simp] theorem walkProduct_cast (N : ℕ) (hN : 1 < N) {g k g' k' : NegativeLevel L N}
-    (p : Walk L N g k) (hg : g = g') (hk : k = k') :
-    walkProduct L hdim h Q N hN (p.cast hg hk) = walkProduct L hdim h Q N hN p := by
-  subst g'; subst k'; rfl
-
-/-- The entire subdivided edge has the exact endpoint transport, not merely a limit. -/
-theorem exists_powerWalk (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
-    (N : ℕ) (hN : 1 < N) {q r : V} (hqr : L.Adj q r) (k : ℕ) (hk : k ≤ N) :
-    ∃ p : Walk L N (powerVertex L N q r 0) (powerVertex L N q r k),
-      walkProduct L hdim h Q N hN p =
-        chartTransport h Q q (levelPoint L hdim N (by omega) (powerVertex L N q r 0))
-          (levelPoint L hdim N (by omega) (powerVertex L N q r k)) := by
-  induction k with
-  | zero => exact ⟨.nil _,by simp [chartTransport]⟩
-  | succ k ih =>
-      obtain ⟨p,hp⟩ := ih (by omega)
-      let t : Walk L N (powerVertex L N q r k)
-          (edgeVertex L N (powerVertex L N q r k) q r) := .cons q r hqr (.nil _)
-      let t' := t.cast rfl (powerVertex_step L N hqr (by omega : k < N))
-      refine ⟨p.append t',?_⟩
-      rw [walkProduct_append,hp]
-      simp only [t',walkProduct_cast,t,walkProduct_cons,walkProduct_nil,mul_one]
-      rw [powerLabel_chart L hdim h Q hf hQ N hN hqr (by omega),chartTransport_trans]
-
-def pureVertex (N : ℕ) (q : V) : NegativeLevel L N :=
-  ⟨(artinGenerator L q)⁻¹ ^ N,by
-    simp only [map_pow,map_inv,height_generator]
-    change N • (-1 : ℤ) = -(N : ℤ)
-    simp⟩
-
-omit [Fintype V] [LinearOrder V] in
-theorem powerVertex_zero (N : ℕ) (q r : V) :
-    powerVertex L N q r 0 = pureVertex L N q := by apply Subtype.ext; simp [powerVertex,pureVertex]
-
-omit [Fintype V] [LinearOrder V] in
-theorem powerVertex_last (N : ℕ) (q r : V) :
-    powerVertex L N q r N = pureVertex L N r := by apply Subtype.ext; simp [powerVertex,pureVertex]
-
-def vertexGauge (v : V) : SU2 := gauge h Q v (vertexPoint v)
-
-theorem levelPoint_power_zero (N : ℕ) (hN : 0 < N) {q r : V} (hqr : L.Adj q r) :
-    levelPoint L hdim N hN (powerVertex L N q r 0) = vertexPoint q := by
-  unfold levelPoint
-  have hh : (powerVertex L N q r 0 : ArtinGroup L) =
-      (artinGenerator L q)⁻¹ ^ N * (artinGenerator L r)⁻¹ ^ 0 := by simp [powerVertex]
-  simp only [hh]
-  exact powersPoint_first L hdim hqr N hN _
-
-theorem levelPoint_power_last (N : ℕ) (hN : 0 < N) {q r : V} (hqr : L.Adj q r) :
-    levelPoint L hdim N hN (powerVertex L N q r N) = vertexPoint r := by
-  unfold levelPoint
-  have hh : (powerVertex L N q r N : ArtinGroup L) =
-      (artinGenerator L q)⁻¹ ^ 0 * (artinGenerator L r)⁻¹ ^ N := by simp [powerVertex]
-  simp only [hh]
-  exact powersPoint_last L hdim hqr N hN _
-
-theorem exists_expanded_edge (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
-    (N : ℕ) (hN : 1 < N) {q r : V} (hqr : L.Adj q r) :
-    ∃ p : Walk L N (pureVertex L N q) (pureVertex L N r),
-      walkProduct L hdim h Q N hN p =
-        (vertexGauge h Q q)⁻¹ * h q r * vertexGauge h Q r := by
-  obtain ⟨p,hp⟩ := exists_powerWalk L hdim h Q hf hQ N hN hqr N le_rfl
-  refine ⟨p.cast (powerVertex_zero L N q r) (powerVertex_last L N q r),?_⟩
-  rw [walkProduct_cast,hp,levelPoint_power_zero L hdim N (by omega) hqr,
-    levelPoint_power_last L hdim N (by omega) hqr]
-  unfold chartTransport vertexGauge
-  rw [gauge_transition L h Q hf hQ (Or.inr hqr)
-    (star_vertex L (Or.inr hqr)) (star_vertex L (Or.inl rfl))]
-  group
-
-omit [Fintype V] [LinearOrder V] in
-def graphWalkProduct {a b : V} : L.Walk a b → SU2
-  | .nil => 1
-  | .cons (u := u) (v := v) _ p => h u v * graphWalkProduct p
-
-theorem exists_expanded_walk (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
-    (N : ℕ) (hN : 1 < N) {q r : V} (w : L.Walk q r) :
-    ∃ p : Walk L N (pureVertex L N q) (pureVertex L N r),
-      walkProduct L hdim h Q N hN p =
-        (vertexGauge h Q q)⁻¹ * graphWalkProduct L h w * vertexGauge h Q r := by
-  induction w with
-  | nil => exact ⟨.nil _,by simp [graphWalkProduct]⟩
-  | @cons q t r hqt w ih =>
-      obtain ⟨p,hp⟩ := exists_expanded_edge L hdim h Q hf hQ N hN hqt
-      obtain ⟨s,hs⟩ := ih
-      refine ⟨p.append s,?_⟩
-      rw [walkProduct_append,hp,hs]
-      simp only [graphWalkProduct]
-      group
-
-omit [Fintype V] [LinearOrder V] in
-theorem graphWalkProduct_eq_zipWith {a b : V} (w : L.Walk a b) :
-    graphWalkProduct L h w = (List.zipWith h w.support w.support.tail).prod := by
-  induction w with
-  | nil => simp [graphWalkProduct]
-  | @cons q t r hqt w ih =>
-      cases w with
-      | nil => simp [graphWalkProduct]
-      | cons htl p => simpa only [graphWalkProduct,SimpleGraph.Walk.support_cons,
-          List.tail_cons,List.zipWith_cons_cons,List.prod_cons] using congrArg (h q t * ·) ih
-
-/-- A nontrivial seed holonomy survives every sufficiently translated level. -/
-theorem exists_nontrivial_level_walk (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
-    (N : ℕ) (hN : 1 < N) {q : V} (w : L.Walk q q)
-    (hw : graphWalkProduct L h w ≠ 1) :
-    ∃ p : Walk L N (pureVertex L N q) (pureVertex L N q),
-      walkProduct L hdim h Q N hN p ≠ 1 := by
-  obtain ⟨p,hp⟩ := exists_expanded_walk L hdim h Q hf hQ N hN w
-  refine ⟨p,?_⟩
-  rw [hp]
-  intro he
-  apply hw
-  calc
-    graphWalkProduct L h w = vertexGauge h Q q *
-      ((vertexGauge h Q q)⁻¹ * graphWalkProduct L h w * vertexGauge h Q q) *
-        (vertexGauge h Q q)⁻¹ := by group
-    _ = 1 := by rw [he]; group
-
-
-namespace Walk
-variable {L} {N : ℕ}
-omit [Fintype V] [LinearOrder V] in
-def eval (a : NegativeLevel L N → V → V → SU2)
-    {g k : NegativeLevel L N} : Walk L N g k → SU2
-  | .nil _ => 1
-  | .cons (g := g) q r _ p => a g q r * eval a p
-end Walk
-
-theorem walkProduct_eq_eval (N : ℕ) (hN : 1 < N) {g k : NegativeLevel L N}
-    (p : Walk L N g k) :
-    walkProduct L hdim h Q N hN p = p.eval (edgeLabel L hdim h Q N hN) := by
-  induction p with
-  | nil => rfl
-  | cons q r hqr p ih => simp only [walkProduct,Walk.eval,ih]
-
-end LevelTransport
-
-namespace SourceTransport
-open LevelTransport AffineTransport
-
-def seedWalk : seedGraph.Walk (.inl (.inl none)) (.inl (.inl none)) :=
-  SimpleGraph.Walk.ofSupport SeedConnection.roseLoop
-    (by simp [SeedConnection.roseLoop]) SeedConnection.roseLoop_edges
-
-theorem seedWalk_support : seedWalk.support = SeedConnection.roseLoop :=
-  SimpleGraph.Walk.support_ofSupport _ _
-
-theorem seedWalk_product_ne : graphWalkProduct seedGraph SeedConnection.connection seedWalk ≠ 1 := by
-  rw [graphWalkProduct_eq_zipWith,seedWalk_support]
-  exact SeedConnection.roseLoop_nontrivial
-
-/-- Actual small flat transport for the fixed source level graph. There are no
-representation, direction-map, local-chart, or nontriviality hypotheses here. -/
-theorem source_small_labels (ε : ℝ) (hε : 0 < ε) :
-    ∃ N : ℕ, 1 < N ∧ ∃ a : NegativeLevel seedGraph N → SeedCell → SeedCell → SU2,
-      (∀ g q r, a (edgeVertex seedGraph N g q r) r q = (a g q r)⁻¹) ∧
-      (∀ g q r, dist (a g q r) 1 < ε) ∧
-      (∀ g₀ g₁ g₂ q₀ r₀ q₁ r₁ q₂ r₂,
-        edgeVertex seedGraph N g₀ q₀ r₀ = g₁ →
-        edgeVertex seedGraph N g₁ q₁ r₁ = g₂ →
-        edgeVertex seedGraph N g₂ q₂ r₂ = g₀ →
-        a g₀ q₀ r₀ * a g₁ q₁ r₁ * a g₂ q₂ r₂ = 1) ∧
-      ∃ (g : NegativeLevel seedGraph N) (p : Walk seedGraph N g g), p.eval a ≠ 1 := by
-  let : LinearOrder SeedCell := LinearOrder.lift' (Fintype.equivFin SeedCell)
-    (Fintype.equivFin SeedCell).injective
-  obtain ⟨Q,hQ⟩ := exists_generalPosition SeedConnection.connection
-  have hf : FlatConnection seedGraph SeedConnection.connection :=
-    fun _ _ _ hab hbc hac => SeedConnection.connection_flat hab hbc hac
-  obtain ⟨M,hM⟩ := edgeLabel_uniform seedGraph SeedConnection.clique_card_le_three
-    SeedConnection.connection Q hQ hε
-  let D := Fintype.card SeedCell + 1
-  let N := max M (8 * D + 2)
-  have hMN : M ≤ N := le_max_left _ _
-  have hNbound : 8 * D + 2 ≤ N := le_max_right _ _
-  have hN : 1 < N := by omega
-  have hD : (0 : ℝ) < D := by dsimp [D]; positivity
-  have hNr : (0 : ℝ) < N := by exact_mod_cast (by omega : 0 < N)
-  have hsize : 8 / (N : ℝ) < 1 / (Fintype.card SeedCell + 1 : ℝ) := by
-    have hBN : (8 : ℝ) * D + 2 ≤ N := by exact_mod_cast hNbound
-    have hDcast : (D : ℝ) = (Fintype.card SeedCell : ℝ) + 1 := by simp [D]
-    rw [← hDcast]
-    apply (div_lt_div_iff₀ hNr hD).mpr
-    nlinarith
-  let a := edgeLabel seedGraph SeedConnection.clique_card_le_three
-    SeedConnection.connection Q N hN
-  refine ⟨N,hN,a,?_,?_,?_,?_⟩
-  · exact edgeLabel_reverse seedGraph SeedConnection.clique_card_le_three
-      SeedConnection.connection Q hf hQ N hN
-  · exact hM N hMN hN
-  · exact edgeLabel_triangle seedGraph SeedConnection.clique_card_le_three
-      SeedConnection.connection Q hf hQ N hN hsize
-  · obtain ⟨p,hp⟩ := exists_nontrivial_level_walk seedGraph SeedConnection.clique_card_le_three
-      SeedConnection.connection Q hf hQ N hN seedWalk seedWalk_product_ne
-    refine ⟨_,p,?_⟩
-    rwa [walkProduct_eq_eval] at hp
-
-end SourceTransport
-end EilenbergGanea
-
-
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.AffineTransport
+import OAI.Topology.EilenbergGanea.SeedConnection
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set Filter Topology MeasureTheory
+open scoped Quaternion ContDiff
+
+namespace EilenbergGanea
+namespace LevelTransport
+open TraceWords AffineTransport
+
+variable {V : Type*} [Fintype V] [LinearOrder V] (L : SimpleGraph V)
+variable (hdim : ∀ s : Finset V, (s : Set V).Pairwise L.Adj → s.card ≤ 3)
+
+def point (g : ArtinGroup L) (hg : 0 < traceLength L (artinNormalTrace L g)) : TrianglePoint V :=
+  ⟨artinDirection L g,by
+    apply mem_triangleSet.mpr
+    refine ⟨(traceCounts L (artinNormalTrace L g)).support,
+      hdim _ (traceCounts_support_clique L _),⟨?_,?_⟩,?_⟩
+    · exact traceDirection_nonneg L _
+    · exact traceDirection_sum L hg
+    · intro v hv
+      simp only [Finsupp.notMem_support_iff] at hv
+      simp [artinDirection,traceDirection,hv]⟩
+
+@[simp] theorem point_val (g : ArtinGroup L) (hg : 0 < traceLength L (artinNormalTrace L g)) :
+    (point L hdim g hg : V → ℝ) = artinDirection L g := rfl
+
+theorem point_mem_support (g : ArtinGroup L) (hg : 0 < traceLength L (artinNormalTrace L g))
+    {v : V} (hv : (point L hdim g hg : V → ℝ) v ≠ 0) :
+    v ∈ (traceCounts L (artinNormalTrace L g)).support := by
+  by_contra h
+  have he := Finsupp.notMem_support_iff.mp h
+  exact hv (by simp [point,artinDirection,traceDirection,he])
+
+theorem point_clique (g : ArtinGroup L) (hg : 0 < traceLength L (artinNormalTrace L g)) :
+    CliquePoint L (point L hdim g hg) := by
+  intro a ha v hv
+  by_cases hav : a = v
+  · exact Or.inl hav
+  · exact Or.inr (traceCounts_support_clique L _
+      (point_mem_support L hdim g hg ha) (point_mem_support L hdim g hg hv) hav)
+
+/-- Both hypotheses refer to actual reduced Artin lengths, not an assumed word metric. -/
+theorem point_step (g : ArtinGroup L) (x : Letter V)
+    (hg : 0 < traceLength L (artinNormalTrace L g))
+    (hgx : 0 < traceLength L (artinNormalTrace L (g * letterValue (artinGenerator L) x)))
+    (N : ℕ) (hNg : N ≤ traceLength L (artinNormalTrace L g))
+    (hNgx : N ≤ traceLength L (artinNormalTrace L (g * letterValue (artinGenerator L) x))) :
+    Compatible L (point L hdim g hg) (point L hdim (g * letterValue (artinGenerator L) x) hgx) ∧
+    dist (point L hdim g hg) (point L hdim (g * letterValue (artinGenerator L) x) hgx) ≤
+      2 / (N+1 : ℝ) := by
+  classical
+  obtain ⟨hs,n,_,hn,hbd⟩ := artinDirection_right_step L g x hg hgx
+  have hNn : N ≤ n := by rcases hn with ⟨he,_⟩ | ⟨he,_⟩ <;> omega
+  constructor
+  · intro v w hv hw
+    by_cases he : v = w
+    · exact Or.inl he
+    · apply Or.inr
+      apply hs _ _ he
+      · rcases hv with hv | hv
+        · exact Finset.mem_union_left _ (point_mem_support L hdim g hg hv)
+        · exact Finset.mem_union_right _ (point_mem_support L hdim _ hgx hv)
+      · rcases hw with hw | hw
+        · exact Finset.mem_union_left _ (point_mem_support L hdim g hg hw)
+        · exact Finset.mem_union_right _ (point_mem_support L hdim _ hgx hw)
+  · have hc : (N : ℝ) + 1 ≤ (n : ℝ) + 1 := by exact_mod_cast Nat.add_le_add_right hNn 1
+    have hd : 0 < (N : ℝ) + 1 := by positivity
+    have hfrac : 2 / (n+1 : ℝ) ≤ 2 / (N+1 : ℝ) :=
+      div_le_div_of_nonneg_left (by norm_num) hd hc
+    apply le_trans ?_ hfrac
+    change dist (artinDirection L g) (artinDirection L (g * letterValue (artinGenerator L) x)) ≤ _
+    apply (dist_pi_le_iff (by positivity)).mpr
+    intro v
+    rw [Real.dist_eq,abs_sub_comm]
+    exact le_trans (Finset.single_le_sum (fun w _ => abs_nonneg (artinDirection L (g * letterValue (artinGenerator L) x) w - artinDirection L g w)) (Finset.mem_univ v)) hbd
+
+
+/-- The integral level downstairs; the action is by left multiplication by the height kernel. -/
+abbrev NegativeLevel (N : ℕ) := {g : ArtinGroup L // (height L g).toAdd = -(N : ℤ)}
+
+omit [Fintype V] [LinearOrder V] in
+theorem negative_level_length (N : ℕ) {g : ArtinGroup L}
+    (hg : (height L g).toAdd = -(N : ℤ)) :
+    N ≤ traceLength L (artinNormalTrace L g) := by
+  have hh := height_abs_le_traceLength L g
+  rw [hg,abs_neg,abs_of_nonneg (Nat.cast_nonneg N)] at hh
+  exact_mod_cast hh
+
+def levelPoint (N : ℕ) (hN : 0 < N) (g : NegativeLevel L N) : TrianglePoint V :=
+  point L hdim g (lt_of_lt_of_le hN (negative_level_length L N g.property))
+
+@[simp] theorem levelPoint_val (N : ℕ) (hN : 0 < N) (g : NegativeLevel L N) :
+    (levelPoint L hdim N hN g : V → ℝ) = artinDirection L g := rfl
+
+def upperVertex (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) (v : V) :
+    NegativeLevel L (N-1) :=
+  ⟨(g : ArtinGroup L) * artinGenerator L v,by
+    rw [map_mul,height_generator]
+    change (height L (g : ArtinGroup L)).toAdd + 1 = _
+    rw [g.property]
+    omega⟩
+
+def edgeVertex (N : ℕ) (g : NegativeLevel L N) (q r : V) : NegativeLevel L N :=
+  ⟨(g : ArtinGroup L) * artinGenerator L q * (artinGenerator L r)⁻¹,by
+    simp only [map_mul,map_inv,height_generator]
+    change (height L (g : ArtinGroup L)).toAdd + 1 + -1 = _
+    rw [g.property]
+    omega⟩
+
+omit [Fintype V] [LinearOrder V] in
+theorem edgeVertex_reverse (N : ℕ) (g : NegativeLevel L N) (q r : V) :
+    edgeVertex L N (edgeVertex L N g q r) r q = g := by
+  apply Subtype.ext
+  change (g : ArtinGroup L) * _ * _ * _ * _ = _
+  group
+
+omit [Fintype V] [LinearOrder V] in
+theorem upperVertex_reverse (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) (q r : V) :
+    upperVertex L N hN (edgeVertex L N g q r) r = upperVertex L N hN g q := by
+  apply Subtype.ext
+  dsimp only [edgeVertex,upperVertex]
+  group
+
+theorem upperPoint_step (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) (v : V) :
+    Compatible L (levelPoint L hdim N (by omega) g)
+      (levelPoint L hdim (N-1) (by omega) (upperVertex L N hN g v)) ∧
+    dist (levelPoint L hdim N (by omega) g)
+      (levelPoint L hdim (N-1) (by omega) (upperVertex L N hN g v)) ≤ 2 / (N : ℝ) := by
+  let hu := upperVertex L N hN g v
+  have hlen := negative_level_length L N g.property
+  have hlen' := negative_level_length L (N-1) hu.property
+  dsimp only [hu,upperVertex] at hlen'
+  have H := point_step L hdim (g : ArtinGroup L) (v,true)
+    (lt_of_lt_of_le (by omega : 0 < N) hlen)
+    (by simpa only [letterValue_positive] using lt_of_lt_of_le (by omega : 0 < N-1) hlen')
+    (N-1) (le_trans (Nat.sub_le N 1) hlen)
+    (by simpa only [letterValue_positive] using hlen')
+  have he : ((N-1 : ℕ) : ℝ) + 1 = (N : ℝ) := by exact_mod_cast (show (N-1 : ℕ) + 1 = N by omega)
+  simpa only [levelPoint,upperVertex,letterValue_positive,he] using H
+
+variable (h : V → V → SU2) (Q : V → ℍ)
+
+def edgeLabel (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) (q r : V) : SU2 :=
+  edgeTransport h Q (levelPoint L hdim N (by omega) g)
+    (levelPoint L hdim (N-1) (by omega) (upperVertex L N hN g q))
+    (levelPoint L hdim N (by omega) (edgeVertex L N g q r))
+
+theorem edgeLabel_reverse (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
+    (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) (q r : V) :
+    edgeLabel L hdim h Q N hN (edgeVertex L N g q r) r q =
+      (edgeLabel L hdim h Q N hN g q r)⁻¹ := by
+  have h₁ := (upperPoint_step L hdim N hN g q).1
+  have h₂ := (upperPoint_step L hdim N hN (edgeVertex L N g q r) r).1
+  rw [upperVertex_reverse] at h₂
+  unfold edgeLabel
+  rw [edgeVertex_reverse,upperVertex_reverse]
+  exact edgeTransport_reverse L h Q hf hQ h₁ (compatible_symm L h₂)
+
+theorem edgeLabel_uniform (hQ : GeneralPosition h Q) {ε : ℝ} (hε : 0 < ε) :
+    ∃ M : ℕ, ∀ N : ℕ, M ≤ N → ∀ hN : 1 < N, ∀ g : NegativeLevel L N, ∀ q r : V,
+      dist (edgeLabel L hdim h Q N hN g q r) 1 < ε := by
+  obtain ⟨δ,hδ,H⟩ := edgeTransport_uniform h Q hQ hε
+  obtain ⟨M,hM⟩ := exists_nat_gt (2 / δ)
+  refine ⟨M,?_⟩
+  intro N hMN hN g q r
+  have hN0 : (0 : ℝ) < N := by exact_mod_cast (by omega : 0 < N)
+  have hbound : 2 / (N : ℝ) < δ := by
+    apply (div_lt_iff₀ hN0).mpr
+    have hMN' : (M : ℝ) ≤ N := by exact_mod_cast hMN
+    have hh := (div_lt_iff₀ hδ).mp hM
+    nlinarith
+  apply H
+  · exact lt_of_le_of_lt (upperPoint_step L hdim N hN g q).2 hbound
+  · have hh := (upperPoint_step L hdim N hN (edgeVertex L N g q r) r).2
+    rw [upperVertex_reverse,dist_comm] at hh
+    exact lt_of_le_of_lt hh hbound
+
+
+omit [LinearOrder V] in
+/-- Six compatible generator pieces around a level triangle lie in one flat chart
+once the three edge images are uniformly short. -/
+theorem triangle_of_short (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
+    (b c d u v w : TrianglePoint V) (δ : ℝ) (hδ : 0 < δ)
+    (hsmall : 4 * δ < 1 / (Fintype.card V + 1 : ℝ))
+    (hb : CliquePoint L b) (hc : CliquePoint L c) (hd : CliquePoint L d)
+    (hu : CliquePoint L u) (hv : CliquePoint L v) (hw : CliquePoint L w)
+    (hbu : Compatible L b u) (huc : Compatible L u c)
+    (hcv : Compatible L c v) (hvd : Compatible L v d)
+    (hdw : Compatible L d w) (hwb : Compatible L w b)
+    (dbu : dist b u ≤ δ) (duc : dist u c ≤ δ)
+    (dcv : dist c v ≤ δ) (dvd : dist v d ≤ δ)
+    (_ddw : dist d w ≤ δ) (dwb : dist w b ≤ δ) :
+    edgeTransport h Q b u c * edgeTransport h Q c v d * edgeTransport h Q d w b = 1 := by
+  have dbc : dist b c ≤ 2 * δ := by linarith [dist_triangle b u c]
+  have dbv : dist b v ≤ 3 * δ := by linarith [dist_triangle b c v]
+  have dbd : dist b d ≤ 4 * δ := by linarith [dist_triangle b v d]
+  have dbw : dist b w ≤ δ := by simpa only [dist_comm] using dwb
+  obtain ⟨a,ha⟩ := exists_large_coordinate b
+  apply edgeTransport_triangle L h Q hf hQ b c d u v w hbu huc hcv hvd hdw hwb
+  · exact nearby_star L ha hb (by simpa using lt_of_lt_of_le hδ (le_of_lt (by linarith : δ < 1 / (Fintype.card V + 1 : ℝ))))
+  · exact nearby_star L ha hc (by linarith)
+  · exact nearby_star L ha hd (lt_of_le_of_lt dbd hsmall)
+  · exact nearby_star L ha hu (by linarith)
+  · exact nearby_star L ha hv (by linarith)
+  · exact nearby_star L ha hw (by linarith)
+
+/-- Every oriented three-edge loop with these actual level endpoints is flat.
+In particular both kinds of sections of a three-cube are covered. -/
+theorem edgeLabel_triangle (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
+    (N : ℕ) (hN : 1 < N) (hsize : 8 / (N : ℝ) < 1 / (Fintype.card V + 1 : ℝ))
+    (g₀ g₁ g₂ : NegativeLevel L N) (q₀ r₀ q₁ r₁ q₂ r₂ : V)
+    (e₀ : edgeVertex L N g₀ q₀ r₀ = g₁)
+    (e₁ : edgeVertex L N g₁ q₁ r₁ = g₂)
+    (e₂ : edgeVertex L N g₂ q₂ r₂ = g₀) :
+    edgeLabel L hdim h Q N hN g₀ q₀ r₀ *
+      edgeLabel L hdim h Q N hN g₁ q₁ r₁ *
+      edgeLabel L hdim h Q N hN g₂ q₂ r₂ = 1 := by
+  have h₀ := upperPoint_step L hdim N hN g₀ q₀
+  have h₁ := upperPoint_step L hdim N hN g₁ q₁
+  have h₂ := upperPoint_step L hdim N hN g₂ q₂
+  have j₀ := upperPoint_step L hdim N hN g₁ r₀
+  have j₁ := upperPoint_step L hdim N hN g₂ r₁
+  have j₂ := upperPoint_step L hdim N hN g₀ r₂
+  rw [← e₀,upperVertex_reverse,e₀] at j₀
+  rw [← e₁,upperVertex_reverse,e₁] at j₁
+  rw [← e₂,upperVertex_reverse,e₂] at j₂
+  unfold edgeLabel
+  rw [e₀,e₁,e₂]
+  apply triangle_of_short L h Q hf hQ _ _ _ _ _ _ (2/(N : ℝ))
+  · positivity
+  · convert hsize using 1; ring
+  · exact point_clique L hdim _ _
+  · exact point_clique L hdim _ _
+  · exact point_clique L hdim _ _
+  · exact point_clique L hdim _ _
+  · exact point_clique L hdim _ _
+  · exact point_clique L hdim _ _
+  · exact h₀.1
+  · exact compatible_symm L j₀.1
+  · exact h₁.1
+  · exact compatible_symm L j₁.1
+  · exact h₂.1
+  · exact compatible_symm L j₂.1
+  · exact h₀.2
+  · simpa only [dist_comm] using j₀.2
+  · exact h₁.2
+  · simpa only [dist_comm] using j₁.2
+  · exact h₂.2
+  · simpa only [dist_comm] using j₂.2
+
+
+/-- A vertex of the simplex is also an honest point of its two-skeleton. -/
+def vertexPoint (v : V) : TrianglePoint V :=
+  ⟨fun w => if v = w then 1 else 0,by
+    apply mem_triangleSet.mpr
+    refine ⟨{v},by simp,⟨?_,?_⟩,?_⟩
+    · intro w; dsimp only; split_ifs <;> norm_num
+    · simp
+    · intro w hw
+      simp only [Finset.mem_singleton] at hw
+      simp [Ne.symm hw]⟩
+
+@[simp] theorem vertexPoint_val (v w : V) :
+    (vertexPoint v : V → ℝ) w = if v = w then 1 else 0 := rfl
+
+theorem star_vertex {a v : V} (hav : Related L a v) : Star L a (vertexPoint v) := by
+  intro w hw
+  have he : v = w := by by_contra hh; exact hw (by simp [vertexPoint,hh])
+  simpa only [← he] using hav
+
+omit [Fintype V] [LinearOrder V] in
+theorem negative_powers_upper {q r : V} (hqr : L.Adj q r) (n m : ℕ) :
+    (artinGenerator L q)⁻¹ ^ (n+1) * (artinGenerator L r)⁻¹ ^ m * artinGenerator L q =
+      (artinGenerator L q)⁻¹ ^ n * (artinGenerator L r)⁻¹ ^ m := by
+  have hc := (adjacent_generators_commute L hqr).inv_right.pow_right m
+  calc
+    _ = (artinGenerator L q)⁻¹ ^ n *
+        ((artinGenerator L q)⁻¹ * ((artinGenerator L r)⁻¹ ^ m * artinGenerator L q)) := by
+      simp only [pow_succ]; group
+    _ = (artinGenerator L q)⁻¹ ^ n *
+        ((artinGenerator L q)⁻¹ * (artinGenerator L q * (artinGenerator L r)⁻¹ ^ m)) := by
+      rw [← hc.eq]
+    _ = _ := by group
+
+/-- Negative commuting powers remain exactly in their two-coordinate face. -/
+theorem powersPoint_star {q r : V} (hqr : L.Adj q r) (n m : ℕ)
+    (hg : 0 < traceLength L (artinNormalTrace L
+      ((artinGenerator L q)⁻¹ ^ n * (artinGenerator L r)⁻¹ ^ m))) :
+    Star L q (point L hdim ((artinGenerator L q)⁻¹ ^ n * (artinGenerator L r)⁻¹ ^ m) hg) := by
+  intro v hv
+  by_cases hqv : q = v
+  · exact Or.inl hqv
+  by_cases hrv : r = v
+  · exact Or.inr (hrv ▸ hqr)
+  exfalso
+  apply hv
+  rw [point_val,direction_negative_commuting_powers L hqr]
+  simp [hqv,hrv]
+
+/-- Constant endpoint in the q-direction, stated with a concrete positive height. -/
+theorem powersPoint_first {q r : V} (hqr : L.Adj q r) (N : ℕ) (hN : 0 < N)
+    (hg : 0 < traceLength L (artinNormalTrace L
+      ((artinGenerator L q)⁻¹ ^ N * (artinGenerator L r)⁻¹ ^ 0))) :
+    point L hdim ((artinGenerator L q)⁻¹ ^ N * (artinGenerator L r)⁻¹ ^ 0) hg =
+      vertexPoint q := by
+  apply Subtype.ext
+  funext v
+  rw [point_val,vertexPoint_val,direction_negative_commuting_powers L hqr]
+  have hn : (N : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hN
+  by_cases hv : q = v <;> simp [hv,hn]
+
+theorem powersPoint_last {q r : V} (hqr : L.Adj q r) (N : ℕ) (hN : 0 < N)
+    (hg : 0 < traceLength L (artinNormalTrace L
+      ((artinGenerator L q)⁻¹ ^ 0 * (artinGenerator L r)⁻¹ ^ N))) :
+    point L hdim ((artinGenerator L q)⁻¹ ^ 0 * (artinGenerator L r)⁻¹ ^ N) hg =
+      vertexPoint r := by
+  apply Subtype.ext
+  funext v
+  rw [point_val,vertexPoint_val,direction_negative_commuting_powers L hqr]
+  have hn : (N : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hN
+  by_cases hv : r = v <;> simp [hv,hn]
+
+def powerVertex (N : ℕ) (q r : V) (j : ℕ) : NegativeLevel L N :=
+  ⟨(artinGenerator L q)⁻¹ ^ (N - min j N) * (artinGenerator L r)⁻¹ ^ min j N,by
+    simp only [map_mul,map_pow,map_inv,height_generator]
+    change (N-min j N : ℕ) • (-1 : ℤ) + (min j N : ℕ) • (-1 : ℤ) = _
+    simp only [nsmul_eq_mul,mul_neg,mul_one]
+    have hh : min j N ≤ N := min_le_right _ _
+    omega⟩
+
+omit [Fintype V] [LinearOrder V] in
+theorem powerVertex_step (N : ℕ) {q r : V} (hqr : L.Adj q r) {j : ℕ} (hj : j < N) :
+    edgeVertex L N (powerVertex L N q r j) q r = powerVertex L N q r (j+1) := by
+  apply Subtype.ext
+  dsimp only [powerVertex,edgeVertex]
+  rw [min_eq_left (by omega : j ≤ N),min_eq_left (by omega : j+1 ≤ N)]
+  have he : N-j = (N-(j+1))+1 := by omega
+  rw [he,negative_powers_upper L hqr,pow_succ]
+  group
+
+omit [Fintype V] [LinearOrder V] in
+theorem powerVertex_upper (N : ℕ) (hN : 1 < N) {q r : V}
+    (hqr : L.Adj q r) {j : ℕ} (hj : j < N) :
+    (upperVertex L N hN (powerVertex L N q r j) q : ArtinGroup L) =
+      (artinGenerator L q)⁻¹ ^ (N-(j+1)) * (artinGenerator L r)⁻¹ ^ j := by
+  dsimp only [powerVertex,upperVertex]
+  rw [min_eq_left (by omega : j ≤ N)]
+  have he : N-j = (N-(j+1))+1 := by omega
+  rw [he,negative_powers_upper L hqr]
+
+theorem powerVertex_star (N : ℕ) (hN : 0 < N) {q r : V} (hqr : L.Adj q r) (j : ℕ) :
+    Star L q (levelPoint L hdim N hN (powerVertex L N q r j)) :=
+  powersPoint_star L hdim hqr _ _ _
+
+theorem powerVertex_upper_star (N : ℕ) (hN : 1 < N) {q r : V}
+    (hqr : L.Adj q r) {j : ℕ} (hj : j < N) :
+    Star L q (levelPoint L hdim (N-1) (by omega)
+      (upperVertex L N hN (powerVertex L N q r j) q)) := by
+  unfold levelPoint
+  generalize_proofs hpos
+  simp only [powerVertex_upper L N hN hqr hj] at hpos ⊢
+  exact powersPoint_star L hdim hqr _ _ _
+
+theorem powerLabel_chart (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
+    (N : ℕ) (hN : 1 < N) {q r : V} (hqr : L.Adj q r) {j : ℕ} (hj : j < N) :
+    edgeLabel L hdim h Q N hN (powerVertex L N q r j) q r =
+      chartTransport h Q q (levelPoint L hdim N (by omega) (powerVertex L N q r j))
+        (levelPoint L hdim N (by omega) (powerVertex L N q r (j+1))) := by
+  have hb := upperPoint_step L hdim N hN (powerVertex L N q r j) q
+  have hc := upperPoint_step L hdim N hN (edgeVertex L N (powerVertex L N q r j) q r) r
+  rw [upperVertex_reverse] at hc
+  unfold edgeLabel
+  rw [edgeTransport_chart L h Q hf hQ hb.1 (compatible_symm L hc.1)
+    (powerVertex_star L hdim N (by omega) hqr j) (powerVertex_upper_star L hdim N hN hqr hj)]
+  · rw [powerVertex_step L N hqr hj]
+  · rw [powerVertex_step L N hqr hj]
+    exact powerVertex_star L hdim N (by omega) hqr (j+1)
+
+
+/-- Directed walks retain the chosen generator pair, including parallel edges. -/
+inductive Walk (N : ℕ) : NegativeLevel L N → NegativeLevel L N → Type _
+  | nil (g : NegativeLevel L N) : Walk N g g
+  | cons {g k : NegativeLevel L N} (q r : V) (hqr : L.Adj q r)
+      (p : Walk N (edgeVertex L N g q r) k) : Walk N g k
+
+namespace Walk
+variable {L} {N : ℕ}
+
+def append {g k l : NegativeLevel L N} : Walk L N g k → Walk L N k l → Walk L N g l
+  | .nil _, p => p
+  | .cons q r hqr p, t => .cons q r hqr (append p t)
+
+def cast {g k g' k' : NegativeLevel L N} (p : Walk L N g k)
+    (hg : g = g') (hk : k = k') : Walk L N g' k' := hg ▸ hk ▸ p
+
+end Walk
+
+def walkProduct (N : ℕ) (hN : 1 < N) {g k : NegativeLevel L N} : Walk L N g k → SU2
+  | .nil _ => 1
+  | .cons (g := g) q r _ p => edgeLabel L hdim h Q N hN g q r * walkProduct N hN p
+
+@[simp] theorem walkProduct_nil (N : ℕ) (hN : 1 < N) (g : NegativeLevel L N) :
+    walkProduct L hdim h Q N hN (.nil g) = 1 := rfl
+
+@[simp] theorem walkProduct_cons (N : ℕ) (hN : 1 < N) {g k : NegativeLevel L N}
+    (q r : V) (hqr : L.Adj q r) (p : Walk L N (edgeVertex L N g q r) k) :
+    walkProduct L hdim h Q N hN (.cons q r hqr p) =
+      edgeLabel L hdim h Q N hN g q r * walkProduct L hdim h Q N hN p := rfl
+
+@[simp] theorem walkProduct_append (N : ℕ) (hN : 1 < N) {g k l : NegativeLevel L N}
+    (p : Walk L N g k) (t : Walk L N k l) :
+    walkProduct L hdim h Q N hN (p.append t) =
+      walkProduct L hdim h Q N hN p * walkProduct L hdim h Q N hN t := by
+  induction p with
+  | nil => simp [Walk.append]
+  | cons q r hqr p ih => simp [Walk.append,ih,mul_assoc]
+
+@[simp] theorem walkProduct_cast (N : ℕ) (hN : 1 < N) {g k g' k' : NegativeLevel L N}
+    (p : Walk L N g k) (hg : g = g') (hk : k = k') :
+    walkProduct L hdim h Q N hN (p.cast hg hk) = walkProduct L hdim h Q N hN p := by
+  subst g'; subst k'; rfl
+
+/-- The entire subdivided edge has the exact endpoint transport, not merely a limit. -/
+theorem exists_powerWalk (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
+    (N : ℕ) (hN : 1 < N) {q r : V} (hqr : L.Adj q r) (k : ℕ) (hk : k ≤ N) :
+    ∃ p : Walk L N (powerVertex L N q r 0) (powerVertex L N q r k),
+      walkProduct L hdim h Q N hN p =
+        chartTransport h Q q (levelPoint L hdim N (by omega) (powerVertex L N q r 0))
+          (levelPoint L hdim N (by omega) (powerVertex L N q r k)) := by
+  induction k with
+  | zero => exact ⟨.nil _,by simp [chartTransport]⟩
+  | succ k ih =>
+      obtain ⟨p,hp⟩ := ih (by omega)
+      let t : Walk L N (powerVertex L N q r k)
+          (edgeVertex L N (powerVertex L N q r k) q r) := .cons q r hqr (.nil _)
+      let t' := t.cast rfl (powerVertex_step L N hqr (by omega : k < N))
+      refine ⟨p.append t',?_⟩
+      rw [walkProduct_append,hp]
+      simp only [t',walkProduct_cast,t,walkProduct_cons,walkProduct_nil,mul_one]
+      rw [powerLabel_chart L hdim h Q hf hQ N hN hqr (by omega),chartTransport_trans]
+
+def pureVertex (N : ℕ) (q : V) : NegativeLevel L N :=
+  ⟨(artinGenerator L q)⁻¹ ^ N,by
+    simp only [map_pow,map_inv,height_generator]
+    change N • (-1 : ℤ) = -(N : ℤ)
+    simp⟩
+
+omit [Fintype V] [LinearOrder V] in
+theorem powerVertex_zero (N : ℕ) (q r : V) :
+    powerVertex L N q r 0 = pureVertex L N q := by apply Subtype.ext; simp [powerVertex,pureVertex]
+
+omit [Fintype V] [LinearOrder V] in
+theorem powerVertex_last (N : ℕ) (q r : V) :
+    powerVertex L N q r N = pureVertex L N r := by apply Subtype.ext; simp [powerVertex,pureVertex]
+
+def vertexGauge (v : V) : SU2 := gauge h Q v (vertexPoint v)
+
+theorem levelPoint_power_zero (N : ℕ) (hN : 0 < N) {q r : V} (hqr : L.Adj q r) :
+    levelPoint L hdim N hN (powerVertex L N q r 0) = vertexPoint q := by
+  unfold levelPoint
+  have hh : (powerVertex L N q r 0 : ArtinGroup L) =
+      (artinGenerator L q)⁻¹ ^ N * (artinGenerator L r)⁻¹ ^ 0 := by simp [powerVertex]
+  simp only [hh]
+  exact powersPoint_first L hdim hqr N hN _
+
+theorem levelPoint_power_last (N : ℕ) (hN : 0 < N) {q r : V} (hqr : L.Adj q r) :
+    levelPoint L hdim N hN (powerVertex L N q r N) = vertexPoint r := by
+  unfold levelPoint
+  have hh : (powerVertex L N q r N : ArtinGroup L) =
+      (artinGenerator L q)⁻¹ ^ 0 * (artinGenerator L r)⁻¹ ^ N := by simp [powerVertex]
+  simp only [hh]
+  exact powersPoint_last L hdim hqr N hN _
+
+theorem exists_expanded_edge (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
+    (N : ℕ) (hN : 1 < N) {q r : V} (hqr : L.Adj q r) :
+    ∃ p : Walk L N (pureVertex L N q) (pureVertex L N r),
+      walkProduct L hdim h Q N hN p =
+        (vertexGauge h Q q)⁻¹ * h q r * vertexGauge h Q r := by
+  obtain ⟨p,hp⟩ := exists_powerWalk L hdim h Q hf hQ N hN hqr N le_rfl
+  refine ⟨p.cast (powerVertex_zero L N q r) (powerVertex_last L N q r),?_⟩
+  rw [walkProduct_cast,hp,levelPoint_power_zero L hdim N (by omega) hqr,
+    levelPoint_power_last L hdim N (by omega) hqr]
+  unfold chartTransport vertexGauge
+  rw [gauge_transition L h Q hf hQ (Or.inr hqr)
+    (star_vertex L (Or.inr hqr)) (star_vertex L (Or.inl rfl))]
+  group
+
+omit [Fintype V] [LinearOrder V] in
+def graphWalkProduct {a b : V} : L.Walk a b → SU2
+  | .nil => 1
+  | .cons (u := u) (v := v) _ p => h u v * graphWalkProduct p
+
+theorem exists_expanded_walk (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
+    (N : ℕ) (hN : 1 < N) {q r : V} (w : L.Walk q r) :
+    ∃ p : Walk L N (pureVertex L N q) (pureVertex L N r),
+      walkProduct L hdim h Q N hN p =
+        (vertexGauge h Q q)⁻¹ * graphWalkProduct L h w * vertexGauge h Q r := by
+  induction w with
+  | nil => exact ⟨.nil _,by simp [graphWalkProduct]⟩
+  | @cons q t r hqt w ih =>
+      obtain ⟨p,hp⟩ := exists_expanded_edge L hdim h Q hf hQ N hN hqt
+      obtain ⟨s,hs⟩ := ih
+      refine ⟨p.append s,?_⟩
+      rw [walkProduct_append,hp,hs]
+      simp only [graphWalkProduct]
+      group
+
+omit [Fintype V] [LinearOrder V] in
+theorem graphWalkProduct_eq_zipWith {a b : V} (w : L.Walk a b) :
+    graphWalkProduct L h w = (List.zipWith h w.support w.support.tail).prod := by
+  induction w with
+  | nil => simp [graphWalkProduct]
+  | @cons q t r hqt w ih =>
+      cases w with
+      | nil => simp [graphWalkProduct]
+      | cons htl p => simpa only [graphWalkProduct,SimpleGraph.Walk.support_cons,
+          List.tail_cons,List.zipWith_cons_cons,List.prod_cons] using congrArg (h q t * ·) ih
+
+/-- A nontrivial seed holonomy survives every sufficiently translated level. -/
+theorem exists_nontrivial_level_walk (hf : FlatConnection L h) (hQ : GeneralPosition h Q)
+    (N : ℕ) (hN : 1 < N) {q : V} (w : L.Walk q q)
+    (hw : graphWalkProduct L h w ≠ 1) :
+    ∃ p : Walk L N (pureVertex L N q) (pureVertex L N q),
+      walkProduct L hdim h Q N hN p ≠ 1 := by
+  obtain ⟨p,hp⟩ := exists_expanded_walk L hdim h Q hf hQ N hN w
+  refine ⟨p,?_⟩
+  rw [hp]
+  intro he
+  apply hw
+  calc
+    graphWalkProduct L h w = vertexGauge h Q q *
+      ((vertexGauge h Q q)⁻¹ * graphWalkProduct L h w * vertexGauge h Q q) *
+        (vertexGauge h Q q)⁻¹ := by group
+    _ = 1 := by rw [he]; group
+
+
+namespace Walk
+variable {L} {N : ℕ}
+omit [Fintype V] [LinearOrder V] in
+def eval (a : NegativeLevel L N → V → V → SU2)
+    {g k : NegativeLevel L N} : Walk L N g k → SU2
+  | .nil _ => 1
+  | .cons (g := g) q r _ p => a g q r * eval a p
+end Walk
+
+theorem walkProduct_eq_eval (N : ℕ) (hN : 1 < N) {g k : NegativeLevel L N}
+    (p : Walk L N g k) :
+    walkProduct L hdim h Q N hN p = p.eval (edgeLabel L hdim h Q N hN) := by
+  induction p with
+  | nil => rfl
+  | cons q r hqr p ih => simp only [walkProduct,Walk.eval,ih]
+
+end LevelTransport
+
+namespace SourceTransport
+open LevelTransport AffineTransport
+
+def seedWalk : seedGraph.Walk (.inl (.inl none)) (.inl (.inl none)) :=
+  SimpleGraph.Walk.ofSupport SeedConnection.roseLoop
+    (by simp [SeedConnection.roseLoop]) SeedConnection.roseLoop_edges
+
+theorem seedWalk_support : seedWalk.support = SeedConnection.roseLoop :=
+  SimpleGraph.Walk.support_ofSupport _ _
+
+theorem seedWalk_product_ne : graphWalkProduct seedGraph SeedConnection.connection seedWalk ≠ 1 := by
+  rw [graphWalkProduct_eq_zipWith,seedWalk_support]
+  exact SeedConnection.roseLoop_nontrivial
+
+/-- Actual small flat transport for the fixed source level graph. There are no
+representation, direction-map, local-chart, or nontriviality hypotheses here. -/
+theorem source_small_labels (ε : ℝ) (hε : 0 < ε) :
+    ∃ N : ℕ, 1 < N ∧ ∃ a : NegativeLevel seedGraph N → SeedCell → SeedCell → SU2,
+      (∀ g q r, a (edgeVertex seedGraph N g q r) r q = (a g q r)⁻¹) ∧
+      (∀ g q r, dist (a g q r) 1 < ε) ∧
+      (∀ g₀ g₁ g₂ q₀ r₀ q₁ r₁ q₂ r₂,
+        edgeVertex seedGraph N g₀ q₀ r₀ = g₁ →
+        edgeVertex seedGraph N g₁ q₁ r₁ = g₂ →
+        edgeVertex seedGraph N g₂ q₂ r₂ = g₀ →
+        a g₀ q₀ r₀ * a g₁ q₁ r₁ * a g₂ q₂ r₂ = 1) ∧
+      ∃ (g : NegativeLevel seedGraph N) (p : Walk seedGraph N g g), p.eval a ≠ 1 := by
+  let : LinearOrder SeedCell := LinearOrder.lift' (Fintype.equivFin SeedCell)
+    (Fintype.equivFin SeedCell).injective
+  obtain ⟨Q,hQ⟩ := exists_generalPosition SeedConnection.connection
+  have hf : FlatConnection seedGraph SeedConnection.connection :=
+    fun _ _ _ hab hbc hac => SeedConnection.connection_flat hab hbc hac
+  obtain ⟨M,hM⟩ := edgeLabel_uniform seedGraph SeedConnection.clique_card_le_three
+    SeedConnection.connection Q hQ hε
+  let D := Fintype.card SeedCell + 1
+  let N := max M (8 * D + 2)
+  have hMN : M ≤ N := le_max_left _ _
+  have hNbound : 8 * D + 2 ≤ N := le_max_right _ _
+  have hN : 1 < N := by omega
+  have hD : (0 : ℝ) < D := by dsimp [D]; positivity
+  have hNr : (0 : ℝ) < N := by exact_mod_cast (by omega : 0 < N)
+  have hsize : 8 / (N : ℝ) < 1 / (Fintype.card SeedCell + 1 : ℝ) := by
+    have hBN : (8 : ℝ) * D + 2 ≤ N := by exact_mod_cast hNbound
+    have hDcast : (D : ℝ) = (Fintype.card SeedCell : ℝ) + 1 := by simp [D]
+    rw [← hDcast]
+    apply (div_lt_div_iff₀ hNr hD).mpr
+    nlinarith
+  let a := edgeLabel seedGraph SeedConnection.clique_card_le_three
+    SeedConnection.connection Q N hN
+  refine ⟨N,hN,a,?_,?_,?_,?_⟩
+  · exact edgeLabel_reverse seedGraph SeedConnection.clique_card_le_three
+      SeedConnection.connection Q hf hQ N hN
+  · exact hM N hMN hN
+  · exact edgeLabel_triangle seedGraph SeedConnection.clique_card_le_three
+      SeedConnection.connection Q hf hQ N hN hsize
+  · obtain ⟨p,hp⟩ := exists_nontrivial_level_walk seedGraph SeedConnection.clique_card_le_three
+      SeedConnection.connection Q hf hQ N hN seedWalk seedWalk_product_ne
+    refine ⟨_,p,?_⟩
+    rwa [walkProduct_eq_eval] at hp
+
+end SourceTransport
+end EilenbergGanea
+
+
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/Main.lean

@@ -1,43 +1,43 @@
-import OAI.Topology.EilenbergGanea.Dimension
-import OAI.Topology.EilenbergGanea.CubicalCover
-import OAI.AlgebraicTopology.CWComplex.Presentation
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set Metric Topology
-
-namespace EilenbergGanea
-universe u
-
-/-- Exclusion is for arbitrary ordinary2-dimensional classifying CW spaces,
-not merely finite presentations or cell complexes with finitely many orbits. -/
-theorem source_no_classifyingSpace_two : ¬ HasClassifyingSpace.{u} SourceGroup 2 := by
-  rintro ⟨X,tX,hTX,cX,hd,hpc,x,⟨χ⟩,E,tE,hcon,p,cov,hs⟩
-  let : TopologicalSpace X := tX
-  let : T2Space X := hTX
-  let : Topology.CWComplex (Set.univ : Set X) := cX
-  let : PathConnectedSpace X := hpc
-  let : TopologicalSpace E := tE
-  let : ContractibleSpace E := hcon
-  obtain ⟨A,a,ha,r,hr,hp,B,hB⟩ := CWPresentation.ordinary_boundary_basis cov hd hs x χ
-  exact source_no_boundary_basis a ha r hr hp B hB
-
-/-- Source Theorem1.1 for the exact103-cell flag triangulation and its actual
-Artin height kernel. The last conjunct is polymorphic in the universe of the
-hypothetical ordinary CW model, with unrestricted numbers of cells. -/
-theorem source_main :
-    Group.FG SourceGroup ∧ Group.ResiduallyFinite SourceGroup ∧
-    cohomologicalDimension SourceGroup = 2 ∧
-    HasClassifyingSpace.{0} SourceGroup 3 ∧
-    ¬ HasClassifyingSpace.{u} SourceGroup 2 :=
-  ⟨source_finitely_generated,source_residuallyFinite,source_cohomologicalDimension,
-    source_hasClassifyingSpace_three,source_no_classifyingSpace_two⟩
-
-end EilenbergGanea
-
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.Dimension
+import OAI.Topology.EilenbergGanea.CubicalCover
+import OAI.AlgebraicTopology.CWComplex.Presentation
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set Metric Topology
+
+namespace EilenbergGanea
+universe u
+
+/-- Exclusion is for arbitrary ordinary2-dimensional classifying CW spaces,
+not merely finite presentations or cell complexes with finitely many orbits. -/
+theorem source_no_classifyingSpace_two : ¬ HasClassifyingSpace.{u} SourceGroup 2 := by
+  rintro ⟨X,tX,hTX,cX,hd,hpc,x,⟨χ⟩,E,tE,hcon,p,cov,hs⟩
+  let : TopologicalSpace X := tX
+  let : T2Space X := hTX
+  let : Topology.CWComplex (Set.univ : Set X) := cX
+  let : PathConnectedSpace X := hpc
+  let : TopologicalSpace E := tE
+  let : ContractibleSpace E := hcon
+  obtain ⟨A,a,ha,r,hr,hp,B,hB⟩ := CWPresentation.ordinary_boundary_basis cov hd hs x χ
+  exact source_no_boundary_basis a ha r hr hp B hB
+
+/-- Source Theorem1.1 for the exact103-cell flag triangulation and its actual
+Artin height kernel. The last conjunct is polymorphic in the universe of the
+hypothetical ordinary CW model, with unrestricted numbers of cells. -/
+theorem source_main :
+    Group.FG SourceGroup ∧ Group.ResiduallyFinite SourceGroup ∧
+    cohomologicalDimension SourceGroup = 2 ∧
+    HasClassifyingSpace.{0} SourceGroup 3 ∧
+    ¬ HasClassifyingSpace.{u} SourceGroup 2 :=
+  ⟨source_finitely_generated,source_residuallyFinite,source_cohomologicalDimension,
+    source_hasClassifyingSpace_three,source_no_classifyingSpace_two⟩
+
+end EilenbergGanea
+
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/Model.lean

@@ -1,61 +1,61 @@
-import OAI.Topology.EilenbergGanea.Seed
-import OAI.GroupTheory.RightAngledArtin.ResidualFiniteness
-import Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension
-import Mathlib.RepresentationTheory.Basic
-import Mathlib.Topology.CWComplex.Classical.Basic
-import Mathlib.Topology.Covering.Basic
-import Mathlib.Topology.Homotopy.Contractible
-import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set Topology
-
-namespace EilenbergGanea
-abbrev SourceGroup := (height seedGraph).ker
-
-/-- Integral, left-module cohomological dimension; no rational replacement. -/
-def cohomologicalDimension (G : Type*) [Group G] : WithBot ℕ∞ :=
-  CategoryTheory.projectiveDimension
-    (ModuleCat.of (MonoidAlgebra ℤ G) (Representation.trivial ℤ G ℤ).asModule)
-
-universe u
-/-- Ordinary K(G,1) of dimension at most n. The cell and space types have no
-finiteness or countability restriction. The contractible surjective covering
-is the usual universal-cover characterization of a classifying space. -/
-def HasClassifyingSpace (G : Type*) [Group G] (n : ℕ) : Prop :=
-  ∃ (X : Type u) (_ : TopologicalSpace X) (_ : T2Space X)
-    (_ : Topology.CWComplex (Set.univ : Set X)),
-    (∀ k : ℕ, n < k → IsEmpty (Topology.CWComplex.cell (Set.univ : Set X) k)) ∧
-    PathConnectedSpace X ∧
-    ∃ x : X, Nonempty (FundamentalGroup X x ≃* G) ∧
-    ∃ (E : Type u) (_ : TopologicalSpace E), ContractibleSpace E ∧
-    ∃ p : E → X, IsCoveringMap p ∧ Function.Surjective p
-
-
-end EilenbergGanea
-
-namespace EilenbergGanea
-/-- The finite-generation conjunct for the exact source height kernel. -/
-theorem source_finitely_generated : Group.FG SourceGroup :=
-  height_kernel_fg seedGraph seed_connected.preconnected seedBase
-
-
-
-end EilenbergGanea
-
-namespace EilenbergGanea
-/-- The second exact conjunct of the source main theorem. -/
-theorem source_residuallyFinite : Group.ResiduallyFinite SourceGroup := by
-  have := artin_residuallyFinite SeedCell seedGraph
-  infer_instance
-
-end EilenbergGanea
-
-
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.Seed
+import OAI.GroupTheory.RightAngledArtin.ResidualFiniteness
+import Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension
+import Mathlib.RepresentationTheory.Basic
+import Mathlib.Topology.CWComplex.Classical.Basic
+import Mathlib.Topology.Covering.Basic
+import Mathlib.Topology.Homotopy.Contractible
+import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set Topology
+
+namespace EilenbergGanea
+abbrev SourceGroup := (height seedGraph).ker
+
+/-- Integral, left-module cohomological dimension; no rational replacement. -/
+def cohomologicalDimension (G : Type*) [Group G] : WithBot ℕ∞ :=
+  CategoryTheory.projectiveDimension
+    (ModuleCat.of (MonoidAlgebra ℤ G) (Representation.trivial ℤ G ℤ).asModule)
+
+universe u
+/-- Ordinary K(G,1) of dimension at most n. The cell and space types have no
+finiteness or countability restriction. The contractible surjective covering
+is the usual universal-cover characterization of a classifying space. -/
+def HasClassifyingSpace (G : Type*) [Group G] (n : ℕ) : Prop :=
+  ∃ (X : Type u) (_ : TopologicalSpace X) (_ : T2Space X)
+    (_ : Topology.CWComplex (Set.univ : Set X)),
+    (∀ k : ℕ, n < k → IsEmpty (Topology.CWComplex.cell (Set.univ : Set X) k)) ∧
+    PathConnectedSpace X ∧
+    ∃ x : X, Nonempty (FundamentalGroup X x ≃* G) ∧
+    ∃ (E : Type u) (_ : TopologicalSpace E), ContractibleSpace E ∧
+    ∃ p : E → X, IsCoveringMap p ∧ Function.Surjective p
+
+
+end EilenbergGanea
+
+namespace EilenbergGanea
+/-- The finite-generation conjunct for the exact source height kernel. -/
+theorem source_finitely_generated : Group.FG SourceGroup :=
+  height_kernel_fg seedGraph seed_connected.preconnected seedBase
+
+
+
+end EilenbergGanea
+
+namespace EilenbergGanea
+/-- The second exact conjunct of the source main theorem. -/
+theorem source_residuallyFinite : Group.ResiduallyFinite SourceGroup := by
+  have := artin_residuallyFinite SeedCell seedGraph
+  infer_instance
+
+end EilenbergGanea
+
+
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/Seed.lean

@@ -1,164 +1,164 @@
-import Mathlib.Tactic
-import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
-import Mathlib.Data.Fintype.Sum
-import Mathlib.LinearAlgebra.Matrix.Notation
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set
-
-namespace EilenbergGanea
-/-- `false` names x; `true` names y. The second Boolean is the positive sign. -/
-def seedWord : Bool → List (Bool × Bool)
-  | false => [(false, true), (false, true), (true, false), (true, false),
-      (true, false), (true, false), (true, false)]
-  | true => [(false, true), (false, true), (true, true), (false, false),
-      (true, true), (false, false), (true, true), (false, false)]
-
-/-- Numbers of sectors in the two coned boundary polygons. -/
-def boundaryLength (r : Bool) : ℕ := 3 * (seedWord r).length
-
-abbrev RoseVertex := Option (Bool × Fin 2)
-abbrev SeedVertex := RoseVertex ⊕ Bool
-abbrev CircleEdge := Bool × Fin 3
-abbrev Sector := (r : Bool) × Fin (boundaryLength r)
-abbrev SeedEdge := CircleEdge ⊕ Sector
-abbrev SeedCell := SeedVertex ⊕ (SeedEdge ⊕ Sector)
-
-def circleStart (e : CircleEdge) : RoseVertex :=
-  ![none, some (e.1, 0), some (e.1, 1)] e.2
-
-def circleEnd (e : CircleEdge) : RoseVertex :=
-  ![some (e.1, 0), some (e.1, 1), none] e.2
-
-def sectorLetter (s : Sector) : Bool × Bool :=
-  (seedWord s.1).get ⟨s.2.val / 3, by
-    have h := s.2.isLt
-    dsimp [boundaryLength] at h
-    omega⟩
-
-def sectorCircleEdge (s : Sector) : CircleEdge :=
-  ((sectorLetter s).1, if (sectorLetter s).2 then
-    ⟨s.2.val % 3, Nat.mod_lt _ (by decide)⟩ else
-    ⟨2 - s.2.val % 3, by omega⟩)
-
-def sectorStart (s : Sector) : RoseVertex :=
-  if (sectorLetter s).2 then circleStart (sectorCircleEdge s)
-  else circleEnd (sectorCircleEdge s)
-
-def nextSector (s : Sector) : Sector :=
-  ⟨s.1, ⟨(s.2.val + 1) % boundaryLength s.1,
-    Nat.mod_lt _ (by have := s.2.isLt; omega)⟩⟩
-
-def edgeVertices : SeedEdge → Finset SeedVertex
-  | .inl e => {.inl (circleStart e), .inl (circleEnd e)}
-  | .inr s => {.inr s.1, .inl (sectorStart s)}
-
-def sectorEdges (s : Sector) : Finset SeedEdge :=
-  {.inl (sectorCircleEdge s), .inr s, .inr (nextSector s)}
-
-def sectorVertices (s : Sector) : Finset SeedVertex :=
-  (sectorEdges s).biUnion edgeVertices
-
-/-- Proper cell-face incidence, including vertices of closed triangles. -/
-def properFace : SeedCell → SeedCell → Prop
-  | .inl v, .inr (.inl e) => v ∈ edgeVertices e
-  | .inl v, .inr (.inr s) => v ∈ sectorVertices s
-  | .inr (.inl e), .inr (.inr s) => e ∈ sectorEdges s
-  | _, _ => False
-
-def seedGraph : SimpleGraph SeedCell where
-  Adj a b := properFace a b ∨ properFace b a
-  symm := ⟨fun _ _ h => h.elim Or.inr Or.inl⟩
-  loopless := ⟨by
-    intro a
-    rcases a with a | a
-    · simp [properFace]
-    · rcases a with a | a <;> simp [properFace]⟩
-
-
-end EilenbergGanea
-
-namespace EilenbergGanea
-
-/-- The vertex of the rose common to its two circles. -/
-def seedBase : SeedCell := .inl (.inl none)
-
-theorem seed_vertex_edge_adj {v : SeedVertex} {e : SeedEdge} (h : v ∈ edgeVertices e) :
-    seedGraph.Adj (.inl v) (.inr (.inl e)) := Or.inl h
-
-theorem seed_edge_sector_adj {e : SeedEdge} {s : Sector} (h : e ∈ sectorEdges s) :
-    seedGraph.Adj (.inr (.inl e)) (.inr (.inr s)) := Or.inl h
-
-theorem rose_vertex_reachable (v : RoseVertex) :
-    seedGraph.Reachable seedBase (.inl (.inl v)) := by
-  rcases v with _ | ⟨b, i⟩
-  · exact .refl _
-  · fin_cases i
-    · have h₁ : seedGraph.Adj seedBase (.inr (.inl (.inl (b, 0)))) := by
-        apply seed_vertex_edge_adj
-        simp [edgeVertices, circleStart, circleEnd]
-      have h₂ : seedGraph.Adj (.inl (.inl (some (b, 0))))
-          (.inr (.inl (.inl (b, 0)))) := by
-        apply seed_vertex_edge_adj
-        simp [edgeVertices, circleStart, circleEnd]
-      exact h₁.reachable.trans h₂.reachable.symm
-    · have h₁ : seedGraph.Adj seedBase (.inr (.inl (.inl (b, 2)))) := by
-        apply seed_vertex_edge_adj
-        simp [edgeVertices, circleStart, circleEnd]
-      have h₂ : seedGraph.Adj (.inl (.inl (some (b, 1))))
-          (.inr (.inl (.inl (b, 2)))) := by
-        apply seed_vertex_edge_adj
-        simp [edgeVertices, circleStart, circleEnd]
-      exact h₁.reachable.trans h₂.reachable.symm
-
-def firstSector (r : Bool) : Sector :=
-  ⟨r, ⟨0, by cases r <;> decide⟩⟩
-
-theorem seed_vertex_reachable (v : SeedVertex) :
-    seedGraph.Reachable seedBase (.inl v) := by
-  rcases v with v | r
-  · exact rose_vertex_reachable v
-  · let s := firstSector r
-    have h₁ : seedGraph.Adj (.inl (.inl (sectorStart s))) (.inr (.inl (.inr s))) := by
-      apply seed_vertex_edge_adj
-      simp [edgeVertices]
-    have h₂ : seedGraph.Adj (.inl (.inr r)) (.inr (.inl (.inr s))) := by
-      apply seed_vertex_edge_adj
-      simp [edgeVertices, s, firstSector]
-    exact ((rose_vertex_reachable (sectorStart s)).trans h₁.reachable).trans h₂.reachable.symm
-
-theorem seed_edge_reachable (e : SeedEdge) :
-    seedGraph.Reachable seedBase (.inr (.inl e)) := by
-  rcases e with e | s
-  · apply (seed_vertex_reachable (.inl (circleStart e))).trans
-    apply SimpleGraph.Adj.reachable
-    apply seed_vertex_edge_adj
-    simp [edgeVertices]
-  · apply (seed_vertex_reachable (.inr s.1)).trans
-    apply SimpleGraph.Adj.reachable
-    apply seed_vertex_edge_adj
-    simp [edgeVertices]
-
-theorem seed_cell_reachable (x : SeedCell) : seedGraph.Reachable seedBase x := by
-  rcases x with v | e
-  · exact seed_vertex_reachable v
-  · rcases e with e | s
-    · exact seed_edge_reachable e
-    · apply (seed_edge_reachable (.inl (sectorCircleEdge s))).trans
-      apply SimpleGraph.Adj.reachable
-      apply seed_edge_sector_adj
-      simp [sectorEdges]
-
-theorem seed_connected : seedGraph.Connected := by
-  exact { preconnected := fun v w =>
-    (seed_cell_reachable v).symm.trans (seed_cell_reachable w), nonempty := ⟨seedBase⟩ }
-
-
-end EilenbergGanea
-
-end
-
-end OAI
+import Mathlib.Tactic
+import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
+import Mathlib.Data.Fintype.Sum
+import Mathlib.LinearAlgebra.Matrix.Notation
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set
+
+namespace EilenbergGanea
+/-- `false` names x; `true` names y. The second Boolean is the positive sign. -/
+def seedWord : Bool → List (Bool × Bool)
+  | false => [(false, true), (false, true), (true, false), (true, false),
+      (true, false), (true, false), (true, false)]
+  | true => [(false, true), (false, true), (true, true), (false, false),
+      (true, true), (false, false), (true, true), (false, false)]
+
+/-- Numbers of sectors in the two coned boundary polygons. -/
+def boundaryLength (r : Bool) : ℕ := 3 * (seedWord r).length
+
+abbrev RoseVertex := Option (Bool × Fin 2)
+abbrev SeedVertex := RoseVertex ⊕ Bool
+abbrev CircleEdge := Bool × Fin 3
+abbrev Sector := (r : Bool) × Fin (boundaryLength r)
+abbrev SeedEdge := CircleEdge ⊕ Sector
+abbrev SeedCell := SeedVertex ⊕ (SeedEdge ⊕ Sector)
+
+def circleStart (e : CircleEdge) : RoseVertex :=
+  ![none, some (e.1, 0), some (e.1, 1)] e.2
+
+def circleEnd (e : CircleEdge) : RoseVertex :=
+  ![some (e.1, 0), some (e.1, 1), none] e.2
+
+def sectorLetter (s : Sector) : Bool × Bool :=
+  (seedWord s.1).get ⟨s.2.val / 3, by
+    have h := s.2.isLt
+    dsimp [boundaryLength] at h
+    omega⟩
+
+def sectorCircleEdge (s : Sector) : CircleEdge :=
+  ((sectorLetter s).1, if (sectorLetter s).2 then
+    ⟨s.2.val % 3, Nat.mod_lt _ (by decide)⟩ else
+    ⟨2 - s.2.val % 3, by omega⟩)
+
+def sectorStart (s : Sector) : RoseVertex :=
+  if (sectorLetter s).2 then circleStart (sectorCircleEdge s)
+  else circleEnd (sectorCircleEdge s)
+
+def nextSector (s : Sector) : Sector :=
+  ⟨s.1, ⟨(s.2.val + 1) % boundaryLength s.1,
+    Nat.mod_lt _ (by have := s.2.isLt; omega)⟩⟩
+
+def edgeVertices : SeedEdge → Finset SeedVertex
+  | .inl e => {.inl (circleStart e), .inl (circleEnd e)}
+  | .inr s => {.inr s.1, .inl (sectorStart s)}
+
+def sectorEdges (s : Sector) : Finset SeedEdge :=
+  {.inl (sectorCircleEdge s), .inr s, .inr (nextSector s)}
+
+def sectorVertices (s : Sector) : Finset SeedVertex :=
+  (sectorEdges s).biUnion edgeVertices
+
+/-- Proper cell-face incidence, including vertices of closed triangles. -/
+def properFace : SeedCell → SeedCell → Prop
+  | .inl v, .inr (.inl e) => v ∈ edgeVertices e
+  | .inl v, .inr (.inr s) => v ∈ sectorVertices s
+  | .inr (.inl e), .inr (.inr s) => e ∈ sectorEdges s
+  | _, _ => False
+
+def seedGraph : SimpleGraph SeedCell where
+  Adj a b := properFace a b ∨ properFace b a
+  symm := ⟨fun _ _ h => h.elim Or.inr Or.inl⟩
+  loopless := ⟨by
+    intro a
+    rcases a with a | a
+    · simp [properFace]
+    · rcases a with a | a <;> simp [properFace]⟩
+
+
+end EilenbergGanea
+
+namespace EilenbergGanea
+
+/-- The vertex of the rose common to its two circles. -/
+def seedBase : SeedCell := .inl (.inl none)
+
+theorem seed_vertex_edge_adj {v : SeedVertex} {e : SeedEdge} (h : v ∈ edgeVertices e) :
+    seedGraph.Adj (.inl v) (.inr (.inl e)) := Or.inl h
+
+theorem seed_edge_sector_adj {e : SeedEdge} {s : Sector} (h : e ∈ sectorEdges s) :
+    seedGraph.Adj (.inr (.inl e)) (.inr (.inr s)) := Or.inl h
+
+theorem rose_vertex_reachable (v : RoseVertex) :
+    seedGraph.Reachable seedBase (.inl (.inl v)) := by
+  rcases v with _ | ⟨b, i⟩
+  · exact .refl _
+  · fin_cases i
+    · have h₁ : seedGraph.Adj seedBase (.inr (.inl (.inl (b, 0)))) := by
+        apply seed_vertex_edge_adj
+        simp [edgeVertices, circleStart, circleEnd]
+      have h₂ : seedGraph.Adj (.inl (.inl (some (b, 0))))
+          (.inr (.inl (.inl (b, 0)))) := by
+        apply seed_vertex_edge_adj
+        simp [edgeVertices, circleStart, circleEnd]
+      exact h₁.reachable.trans h₂.reachable.symm
+    · have h₁ : seedGraph.Adj seedBase (.inr (.inl (.inl (b, 2)))) := by
+        apply seed_vertex_edge_adj
+        simp [edgeVertices, circleStart, circleEnd]
+      have h₂ : seedGraph.Adj (.inl (.inl (some (b, 1))))
+          (.inr (.inl (.inl (b, 2)))) := by
+        apply seed_vertex_edge_adj
+        simp [edgeVertices, circleStart, circleEnd]
+      exact h₁.reachable.trans h₂.reachable.symm
+
+def firstSector (r : Bool) : Sector :=
+  ⟨r, ⟨0, by cases r <;> decide⟩⟩
+
+theorem seed_vertex_reachable (v : SeedVertex) :
+    seedGraph.Reachable seedBase (.inl v) := by
+  rcases v with v | r
+  · exact rose_vertex_reachable v
+  · let s := firstSector r
+    have h₁ : seedGraph.Adj (.inl (.inl (sectorStart s))) (.inr (.inl (.inr s))) := by
+      apply seed_vertex_edge_adj
+      simp [edgeVertices]
+    have h₂ : seedGraph.Adj (.inl (.inr r)) (.inr (.inl (.inr s))) := by
+      apply seed_vertex_edge_adj
+      simp [edgeVertices, s, firstSector]
+    exact ((rose_vertex_reachable (sectorStart s)).trans h₁.reachable).trans h₂.reachable.symm
+
+theorem seed_edge_reachable (e : SeedEdge) :
+    seedGraph.Reachable seedBase (.inr (.inl e)) := by
+  rcases e with e | s
+  · apply (seed_vertex_reachable (.inl (circleStart e))).trans
+    apply SimpleGraph.Adj.reachable
+    apply seed_vertex_edge_adj
+    simp [edgeVertices]
+  · apply (seed_vertex_reachable (.inr s.1)).trans
+    apply SimpleGraph.Adj.reachable
+    apply seed_vertex_edge_adj
+    simp [edgeVertices]
+
+theorem seed_cell_reachable (x : SeedCell) : seedGraph.Reachable seedBase x := by
+  rcases x with v | e
+  · exact seed_vertex_reachable v
+  · rcases e with e | s
+    · exact seed_edge_reachable e
+    · apply (seed_edge_reachable (.inl (sectorCircleEdge s))).trans
+      apply SimpleGraph.Adj.reachable
+      apply seed_edge_sector_adj
+      simp [sectorEdges]
+
+theorem seed_connected : seedGraph.Connected := by
+  exact { preconnected := fun v w =>
+    (seed_cell_reachable v).symm.trans (seed_cell_reachable w), nonempty := ⟨seedBase⟩ }
+
+
+end EilenbergGanea
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/SeedConnection.lean

@@ -1,277 +1,277 @@
-import OAI.Topology.EilenbergGanea.SeedHolonomy
-import OAI.GroupTheory.RightAngledArtin.Basic
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set Filter Topology MeasureTheory
-open scoped Quaternion ContDiff
-
-namespace EilenbergGanea
-namespace SeedConnection
-
-def circleWord (e : CircleEdge) : FreeGroup Bool := if e.2 = 0 then FreeGroup.of e.1 else 1
-
-def boundaryEdgeWord (s : Sector) : FreeGroup Bool :=
-  if (sectorLetter s).2 then circleWord (sectorCircleEdge s) else
-    (circleWord (sectorCircleEdge s))⁻¹
-
-def boundaryWords (r : Bool) : List (FreeGroup Bool) :=
-  List.ofFn (fun i : Fin (boundaryLength r) => boundaryEdgeWord ⟨r,i⟩)
-
-theorem boundaryWords_prod : ∀ r, (boundaryWords r).prod = seedRelator r := by
-  decide
-
-theorem sector_end : ∀ s : Sector, sectorStart (nextSector s) =
-    if (sectorLetter s).2 then circleEnd (sectorCircleEdge s)
-    else circleStart (sectorCircleEdge s) := by decide
-
-theorem circle_distinct : ∀ e : CircleEdge, circleStart e ≠ circleEnd e := by decide
-
-theorem sector_distinct : ∀ s : Sector, sectorStart s ≠ sectorStart (nextSector s) := by decide
-
-def circleLabel (e : CircleEdge) : SeedSU2 := FreeGroup.lift seedLetterUnit (circleWord e)
-
-def radialLabel (s : Sector) : SeedSU2 :=
-  FreeGroup.lift seedLetterUnit ((boundaryWords s.1).take s.2.val).prod
-
-@[simp] theorem boundaryWords_length (r : Bool) : (boundaryWords r).length = boundaryLength r :=
-  List.length_ofFn
-
-@[simp] theorem boundaryWords_get (s : Sector) :
-    (boundaryWords s.1)[s.2.val]'(by simp) = boundaryEdgeWord s := by
-  simp [boundaryWords]
-
-theorem radial_step (s : Sector) : radialLabel (nextSector s) = radialLabel s *
-    (if (sectorLetter s).2 then circleLabel (sectorCircleEdge s)
-      else (circleLabel (sectorCircleEdge s))⁻¹) := by
-  have hs : s.2.val < (boundaryWords s.1).length := by simp
-  have he := congrArg (FreeGroup.lift seedLetterUnit) (congrArg List.prod
-    (List.take_succ_eq_append_getElem hs))
-  rw [List.prod_append,List.prod_singleton,map_mul,boundaryWords_get] at he
-  have hf : FreeGroup.lift seedLetterUnit (boundaryEdgeWord s) =
-      if (sectorLetter s).2 then circleLabel (sectorCircleEdge s)
-      else (circleLabel (sectorCircleEdge s))⁻¹ := by
-    unfold boundaryEdgeWord circleLabel
-    split <;> simp
-  rw [hf] at he
-  change radialLabel (nextSector s) =
-    FreeGroup.lift seedLetterUnit ((boundaryWords s.1).take s.2.val).prod * _
-  rw [← he]
-  unfold radialLabel nextSector
-  dsimp only
-  by_cases hn : s.2.val + 1 < boundaryLength s.1
-  · rw [Nat.mod_eq_of_lt hn]
-  · have hn' : s.2.val + 1 = boundaryLength s.1 := by have hh := s.2.isLt; omega
-    rw [hn',Nat.mod_self,List.take_zero,List.prod_nil,map_one,
-      ← boundaryWords_length,List.take_length,boundaryWords_prod,seedRelator_unit]
-
-def edgeAnchor : SeedEdge → SeedVertex
-  | .inl e => .inl (circleStart e)
-  | .inr s => .inr s.1
-
-def edgeLocal : SeedEdge → SeedVertex → SeedSU2
-  | .inl e, v => if v = .inl (circleStart e) then 1 else circleLabel e
-  | .inr s, v => if v = .inr s.1 then 1 else radialLabel s
-
-def faceLocal (s : Sector) (v : SeedVertex) : SeedSU2 :=
-  if v = .inr s.1 then 1 else
-  if v = .inl (sectorStart s) then radialLabel s else radialLabel (nextSector s)
-
-theorem edgeLocal_anchor (e : SeedEdge) : edgeLocal e (edgeAnchor e) = 1 := by
-  cases e <;> simp [edgeLocal,edgeAnchor]
-
-theorem nextSector_first (s : Sector) : (nextSector s).1 = s.1 := rfl
-
-theorem faceLocal_incidence (s : Sector) (e : SeedEdge) (he : e ∈ sectorEdges s)
-    (v : SeedVertex) (hv : v ∈ edgeVertices e) :
-    faceLocal s v = faceLocal s (edgeAnchor e) * edgeLocal e v := by
-  simp only [sectorEdges,Finset.mem_insert,Finset.mem_singleton] at he
-  rcases he with rfl | rfl | rfl
-  · simp only [edgeVertices,Finset.mem_insert,Finset.mem_singleton] at hv
-    rcases hv with rfl | rfl
-    · simp [edgeAnchor,edgeLocal]
-    · have hd := circle_distinct (sectorCircleEdge s)
-      have hr := radial_step s
-      cases hb : (sectorLetter s).2
-      · simp only [hb,Bool.false_eq_true,↓reduceIte] at hr
-        simp only [sectorStart,hb,Bool.false_eq_true,↓reduceIte,
-          edgeAnchor,edgeLocal,faceLocal,Sum.inl_ne_inr,
-          Sum.inl.injEq,Ne.symm hd,hd]
-        rw [hr]
-        group
-      · simp only [hb,↓reduceIte] at hr
-        simp only [sectorStart,hb,↓reduceIte,edgeAnchor,edgeLocal,faceLocal,
-          Sum.inl_ne_inr,Sum.inl.injEq,Ne.symm hd]
-        exact hr
-  · simp only [edgeVertices,Finset.mem_insert,Finset.mem_singleton] at hv
-    rcases hv with rfl | rfl <;> simp [faceLocal,edgeLocal,edgeAnchor]
-  · simp only [edgeVertices,Finset.mem_insert,Finset.mem_singleton] at hv
-    rcases hv with rfl | rfl
-    · simp [faceLocal,edgeLocal,edgeAnchor,nextSector_first]
-    · simp [faceLocal,edgeLocal,edgeAnchor,nextSector_first,Ne.symm (sector_distinct s)]
-
-def cellRank : SeedCell → ℕ
-  | .inl _ => 0
-  | .inr (.inl _) => 1
-  | .inr (.inr _) => 2
-
-theorem properFace_rank {a b : SeedCell} (h : properFace a b) : cellRank a < cellRank b := by
-  rcases a with a | (a | a) <;> rcases b with b | (b | b) <;>
-    simp only [properFace,cellRank] at * <;> omega
-
-theorem properFace_trans {a b c : SeedCell} (hab : properFace a b) (hbc : properFace b c) :
-    properFace a c := by
-  rcases a with a | (a | a) <;> rcases b with b | (b | b) <;>
-    rcases c with c | (c | c) <;> simp only [properFace] at *
-  exact Finset.mem_biUnion.mpr ⟨b,hbc,hab⟩
-
-def upLabel : SeedCell → SeedCell → SeedSU2
-  | .inl v, .inr (.inl e) => (edgeLocal e v)⁻¹
-  | .inl v, .inr (.inr s) => (faceLocal s v)⁻¹
-  | .inr (.inl e), .inr (.inr s) => (faceLocal s (edgeAnchor e))⁻¹
-  | _, _ => 1
-
-@[simp] theorem upLabel_diag (a : SeedCell) : upLabel a a = 1 := by
-  rcases a with a | (a | a) <;> rfl
-
-theorem upLabel_same_rank {a b : SeedCell} (hab : cellRank a = cellRank b) :
-    upLabel a b = 1 := by
-  rcases a with a | (a | a) <;> rcases b with b | (b | b) <;>
-    simp [upLabel,cellRank] at *
-
-theorem upLabel_trans {a b c : SeedCell} (hab : a = b ∨ properFace a b)
-    (hbc : b = c ∨ properFace b c) : upLabel a b * upLabel b c = upLabel a c := by
-  rcases hab with rfl | hab
-  · simp
-  rcases hbc with rfl | hbc
-  · simp
-  rcases a with a | (a | a) <;> rcases b with b | (b | b) <;>
-    rcases c with c | (c | c) <;> simp only [properFace] at hab hbc
-  simp only [upLabel,← mul_inv_rev]
-  exact congrArg Inv.inv (faceLocal_incidence c b hbc a hab).symm
-
-def Related (a b : SeedCell) : Prop := a = b ∨ seedGraph.Adj a b
-
-theorem related_symm {a b : SeedCell} : Related a b → Related b a := by
-  rintro (rfl | h)
-  · exact Or.inl rfl
-  · exact Or.inr h.symm
-
-theorem related_forward {a b : SeedCell} (h : Related a b) (hr : cellRank a ≤ cellRank b) :
-    a = b ∨ properFace a b := by
-  rcases h with h | h
-  · exact Or.inl h
-  rcases h with h | h
-  · exact Or.inr h
-  · have hd := properFace_rank h
-    omega
-
-def connection (a b : SeedCell) : SeedSU2 :=
-  if cellRank a ≤ cellRank b then upLabel a b else (upLabel b a)⁻¹
-
-@[simp] theorem connection_diag (a : SeedCell) : connection a a = 1 := by
-  simp [connection]
-
-theorem connection_reverse (a b : SeedCell) : connection b a = (connection a b)⁻¹ := by
-  by_cases hab : cellRank a ≤ cellRank b
-  · by_cases hba : cellRank b ≤ cellRank a
-    · have he := le_antisymm hab hba
-      simp [connection,hab,hba,upLabel_same_rank he,upLabel_same_rank he.symm]
-    · simp [connection,hab,hba]
-  · have hba : cellRank b ≤ cellRank a := (le_total _ _).resolve_left hab
-    simp [connection,hab,hba]
-
-theorem connection_flat_sorted {a b c : SeedCell} (hab : Related a b) (hbc : Related b c)
-    (hrab : cellRank a ≤ cellRank b) (hrbc : cellRank b ≤ cellRank c) :
-    connection a b * connection b c = connection a c := by
-  simp only [connection,hrab,hrbc,le_trans hrab hrbc,↓reduceIte]
-  exact upLabel_trans (related_forward hab hrab) (related_forward hbc hrbc)
-
-theorem connection_flat {a b c : SeedCell} (hab : Related a b) (hbc : Related b c)
-    (hac : Related a c) : connection a b * connection b c = connection a c := by
-  by_cases hrab : cellRank a ≤ cellRank b
-  · by_cases hrbc : cellRank b ≤ cellRank c
-    · exact connection_flat_sorted hab hbc hrab hrbc
-    · have hrcb := le_of_lt (lt_of_not_ge hrbc)
-      by_cases hrac : cellRank a ≤ cellRank c
-      · have H := connection_flat_sorted hac (related_symm hbc) hrac hrcb
-        rw [connection_reverse b c] at H
-        rw [← H]
-        group
-      · have hrca := le_of_lt (lt_of_not_ge hrac)
-        have H := connection_flat_sorted (related_symm hac) hab hrca hrab
-        rw [connection_reverse a c,connection_reverse b c] at H
-        calc
-          connection a b * connection b c =
-              connection a c * ((connection a c)⁻¹ * connection a b) * connection b c := by group
-          _ = connection a c * (connection b c)⁻¹ * connection b c := by rw [H]
-          _ = connection a c := by group
-  · have hrba := le_of_lt (lt_of_not_ge hrab)
-    by_cases hrac : cellRank a ≤ cellRank c
-    · have H := connection_flat_sorted (related_symm hab) hac hrba hrac
-      rw [connection_reverse a b] at H
-      rw [← H]
-      group
-    · have hrca := le_of_lt (lt_of_not_ge hrac)
-      by_cases hrbc : cellRank b ≤ cellRank c
-      · have H := connection_flat_sorted hbc (related_symm hac) hrbc hrca
-        rw [connection_reverse a c,connection_reverse a b] at H
-        calc
-          connection a b * connection b c =
-              connection a b * (connection b c * (connection a c)⁻¹) * connection a c := by group
-          _ = connection a b * (connection a b)⁻¹ * connection a c := by rw [H]
-          _ = connection a c := by group
-      · have hrcb := le_of_lt (lt_of_not_ge hrbc)
-        have H := connection_flat_sorted (related_symm hbc) (related_symm hab) hrcb hrba
-        rw [connection_reverse b c,connection_reverse a b,connection_reverse a c] at H
-        have H' := congrArg Inv.inv H
-        simpa only [mul_inv_rev,inv_inv] using H'
-
-def rankFin (a : SeedCell) : Fin 3 := ⟨cellRank a,by
-  rcases a with a | (a | a) <;> norm_num [cellRank]⟩
-
-theorem adjacent_rank_ne {a b : SeedCell} (h : seedGraph.Adj a b) : rankFin a ≠ rankFin b := by
-  intro he
-  have he' := congrArg Fin.val he
-  change cellRank a = cellRank b at he'
-  rcases h with h | h <;> have hh := properFace_rank h <;> omega
-
-theorem clique_card_le_three (s : Finset SeedCell) (hs : (s : Set SeedCell).Pairwise seedGraph.Adj) :
-    s.card ≤ 3 := by
-  have hi : Set.InjOn rankFin (s : Set SeedCell) := by
-    intro a ha b hb he
-    by_contra hn
-    exact adjacent_rank_ne (hs ha hb hn) he
-  have hc := Finset.card_le_card (Finset.subset_univ (s.image rankFin))
-  rwa [Finset.card_image_of_injOn hi,Finset.card_univ,Fintype.card_fin] at hc
-
-/-- The subdivided x-circle in the exact 103-cell order graph. -/
-def roseLoop : List SeedCell :=
-  [.inl (.inl none), .inr (.inl (.inl (false,0))),
-    .inl (.inl (some (false,0))), .inr (.inl (.inl (false,1))),
-    .inl (.inl (some (false,1))), .inr (.inl (.inl (false,2))), .inl (.inl none)]
-
-theorem roseLoop_edges : roseLoop.IsChain seedGraph.Adj := by
-  simp [roseLoop,List.isChain_cons_cons,seedGraph,properFace,edgeVertices,circleStart,circleEnd]
-
-theorem roseLoop_holonomy : (List.zipWith connection roseLoop roseLoop.tail).prod = seedXUnit := by
-  simp [roseLoop,connection,cellRank,upLabel,edgeLocal,circleLabel,circleWord,
-    circleStart,seedLetterUnit]
-
-theorem roseLoop_nontrivial : (List.zipWith connection roseLoop roseLoop.tail).prod ≠ 1 := by
-  rw [roseLoop_holonomy]
-  intro h
-  have hh := congrArg (fun q : SeedSU2 => (q : ℍ).re) h
-  norm_num [seedXUnit,seedXQuaternion] at hh
-
-end SeedConnection
-
-
-
-end EilenbergGanea
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.SeedHolonomy
+import OAI.GroupTheory.RightAngledArtin.Basic
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set Filter Topology MeasureTheory
+open scoped Quaternion ContDiff
+
+namespace EilenbergGanea
+namespace SeedConnection
+
+def circleWord (e : CircleEdge) : FreeGroup Bool := if e.2 = 0 then FreeGroup.of e.1 else 1
+
+def boundaryEdgeWord (s : Sector) : FreeGroup Bool :=
+  if (sectorLetter s).2 then circleWord (sectorCircleEdge s) else
+    (circleWord (sectorCircleEdge s))⁻¹
+
+def boundaryWords (r : Bool) : List (FreeGroup Bool) :=
+  List.ofFn (fun i : Fin (boundaryLength r) => boundaryEdgeWord ⟨r,i⟩)
+
+theorem boundaryWords_prod : ∀ r, (boundaryWords r).prod = seedRelator r := by
+  decide
+
+theorem sector_end : ∀ s : Sector, sectorStart (nextSector s) =
+    if (sectorLetter s).2 then circleEnd (sectorCircleEdge s)
+    else circleStart (sectorCircleEdge s) := by decide
+
+theorem circle_distinct : ∀ e : CircleEdge, circleStart e ≠ circleEnd e := by decide
+
+theorem sector_distinct : ∀ s : Sector, sectorStart s ≠ sectorStart (nextSector s) := by decide
+
+def circleLabel (e : CircleEdge) : SeedSU2 := FreeGroup.lift seedLetterUnit (circleWord e)
+
+def radialLabel (s : Sector) : SeedSU2 :=
+  FreeGroup.lift seedLetterUnit ((boundaryWords s.1).take s.2.val).prod
+
+@[simp] theorem boundaryWords_length (r : Bool) : (boundaryWords r).length = boundaryLength r :=
+  List.length_ofFn
+
+@[simp] theorem boundaryWords_get (s : Sector) :
+    (boundaryWords s.1)[s.2.val]'(by simp) = boundaryEdgeWord s := by
+  simp [boundaryWords]
+
+theorem radial_step (s : Sector) : radialLabel (nextSector s) = radialLabel s *
+    (if (sectorLetter s).2 then circleLabel (sectorCircleEdge s)
+      else (circleLabel (sectorCircleEdge s))⁻¹) := by
+  have hs : s.2.val < (boundaryWords s.1).length := by simp
+  have he := congrArg (FreeGroup.lift seedLetterUnit) (congrArg List.prod
+    (List.take_succ_eq_append_getElem hs))
+  rw [List.prod_append,List.prod_singleton,map_mul,boundaryWords_get] at he
+  have hf : FreeGroup.lift seedLetterUnit (boundaryEdgeWord s) =
+      if (sectorLetter s).2 then circleLabel (sectorCircleEdge s)
+      else (circleLabel (sectorCircleEdge s))⁻¹ := by
+    unfold boundaryEdgeWord circleLabel
+    split <;> simp
+  rw [hf] at he
+  change radialLabel (nextSector s) =
+    FreeGroup.lift seedLetterUnit ((boundaryWords s.1).take s.2.val).prod * _
+  rw [← he]
+  unfold radialLabel nextSector
+  dsimp only
+  by_cases hn : s.2.val + 1 < boundaryLength s.1
+  · rw [Nat.mod_eq_of_lt hn]
+  · have hn' : s.2.val + 1 = boundaryLength s.1 := by have hh := s.2.isLt; omega
+    rw [hn',Nat.mod_self,List.take_zero,List.prod_nil,map_one,
+      ← boundaryWords_length,List.take_length,boundaryWords_prod,seedRelator_unit]
+
+def edgeAnchor : SeedEdge → SeedVertex
+  | .inl e => .inl (circleStart e)
+  | .inr s => .inr s.1
+
+def edgeLocal : SeedEdge → SeedVertex → SeedSU2
+  | .inl e, v => if v = .inl (circleStart e) then 1 else circleLabel e
+  | .inr s, v => if v = .inr s.1 then 1 else radialLabel s
+
+def faceLocal (s : Sector) (v : SeedVertex) : SeedSU2 :=
+  if v = .inr s.1 then 1 else
+  if v = .inl (sectorStart s) then radialLabel s else radialLabel (nextSector s)
+
+theorem edgeLocal_anchor (e : SeedEdge) : edgeLocal e (edgeAnchor e) = 1 := by
+  cases e <;> simp [edgeLocal,edgeAnchor]
+
+theorem nextSector_first (s : Sector) : (nextSector s).1 = s.1 := rfl
+
+theorem faceLocal_incidence (s : Sector) (e : SeedEdge) (he : e ∈ sectorEdges s)
+    (v : SeedVertex) (hv : v ∈ edgeVertices e) :
+    faceLocal s v = faceLocal s (edgeAnchor e) * edgeLocal e v := by
+  simp only [sectorEdges,Finset.mem_insert,Finset.mem_singleton] at he
+  rcases he with rfl | rfl | rfl
+  · simp only [edgeVertices,Finset.mem_insert,Finset.mem_singleton] at hv
+    rcases hv with rfl | rfl
+    · simp [edgeAnchor,edgeLocal]
+    · have hd := circle_distinct (sectorCircleEdge s)
+      have hr := radial_step s
+      cases hb : (sectorLetter s).2
+      · simp only [hb,Bool.false_eq_true,↓reduceIte] at hr
+        simp only [sectorStart,hb,Bool.false_eq_true,↓reduceIte,
+          edgeAnchor,edgeLocal,faceLocal,Sum.inl_ne_inr,
+          Sum.inl.injEq,Ne.symm hd,hd]
+        rw [hr]
+        group
+      · simp only [hb,↓reduceIte] at hr
+        simp only [sectorStart,hb,↓reduceIte,edgeAnchor,edgeLocal,faceLocal,
+          Sum.inl_ne_inr,Sum.inl.injEq,Ne.symm hd]
+        exact hr
+  · simp only [edgeVertices,Finset.mem_insert,Finset.mem_singleton] at hv
+    rcases hv with rfl | rfl <;> simp [faceLocal,edgeLocal,edgeAnchor]
+  · simp only [edgeVertices,Finset.mem_insert,Finset.mem_singleton] at hv
+    rcases hv with rfl | rfl
+    · simp [faceLocal,edgeLocal,edgeAnchor,nextSector_first]
+    · simp [faceLocal,edgeLocal,edgeAnchor,nextSector_first,Ne.symm (sector_distinct s)]
+
+def cellRank : SeedCell → ℕ
+  | .inl _ => 0
+  | .inr (.inl _) => 1
+  | .inr (.inr _) => 2
+
+theorem properFace_rank {a b : SeedCell} (h : properFace a b) : cellRank a < cellRank b := by
+  rcases a with a | (a | a) <;> rcases b with b | (b | b) <;>
+    simp only [properFace,cellRank] at * <;> omega
+
+theorem properFace_trans {a b c : SeedCell} (hab : properFace a b) (hbc : properFace b c) :
+    properFace a c := by
+  rcases a with a | (a | a) <;> rcases b with b | (b | b) <;>
+    rcases c with c | (c | c) <;> simp only [properFace] at *
+  exact Finset.mem_biUnion.mpr ⟨b,hbc,hab⟩
+
+def upLabel : SeedCell → SeedCell → SeedSU2
+  | .inl v, .inr (.inl e) => (edgeLocal e v)⁻¹
+  | .inl v, .inr (.inr s) => (faceLocal s v)⁻¹
+  | .inr (.inl e), .inr (.inr s) => (faceLocal s (edgeAnchor e))⁻¹
+  | _, _ => 1
+
+@[simp] theorem upLabel_diag (a : SeedCell) : upLabel a a = 1 := by
+  rcases a with a | (a | a) <;> rfl
+
+theorem upLabel_same_rank {a b : SeedCell} (hab : cellRank a = cellRank b) :
+    upLabel a b = 1 := by
+  rcases a with a | (a | a) <;> rcases b with b | (b | b) <;>
+    simp [upLabel,cellRank] at *
+
+theorem upLabel_trans {a b c : SeedCell} (hab : a = b ∨ properFace a b)
+    (hbc : b = c ∨ properFace b c) : upLabel a b * upLabel b c = upLabel a c := by
+  rcases hab with rfl | hab
+  · simp
+  rcases hbc with rfl | hbc
+  · simp
+  rcases a with a | (a | a) <;> rcases b with b | (b | b) <;>
+    rcases c with c | (c | c) <;> simp only [properFace] at hab hbc
+  simp only [upLabel,← mul_inv_rev]
+  exact congrArg Inv.inv (faceLocal_incidence c b hbc a hab).symm
+
+def Related (a b : SeedCell) : Prop := a = b ∨ seedGraph.Adj a b
+
+theorem related_symm {a b : SeedCell} : Related a b → Related b a := by
+  rintro (rfl | h)
+  · exact Or.inl rfl
+  · exact Or.inr h.symm
+
+theorem related_forward {a b : SeedCell} (h : Related a b) (hr : cellRank a ≤ cellRank b) :
+    a = b ∨ properFace a b := by
+  rcases h with h | h
+  · exact Or.inl h
+  rcases h with h | h
+  · exact Or.inr h
+  · have hd := properFace_rank h
+    omega
+
+def connection (a b : SeedCell) : SeedSU2 :=
+  if cellRank a ≤ cellRank b then upLabel a b else (upLabel b a)⁻¹
+
+@[simp] theorem connection_diag (a : SeedCell) : connection a a = 1 := by
+  simp [connection]
+
+theorem connection_reverse (a b : SeedCell) : connection b a = (connection a b)⁻¹ := by
+  by_cases hab : cellRank a ≤ cellRank b
+  · by_cases hba : cellRank b ≤ cellRank a
+    · have he := le_antisymm hab hba
+      simp [connection,hab,hba,upLabel_same_rank he,upLabel_same_rank he.symm]
+    · simp [connection,hab,hba]
+  · have hba : cellRank b ≤ cellRank a := (le_total _ _).resolve_left hab
+    simp [connection,hab,hba]
+
+theorem connection_flat_sorted {a b c : SeedCell} (hab : Related a b) (hbc : Related b c)
+    (hrab : cellRank a ≤ cellRank b) (hrbc : cellRank b ≤ cellRank c) :
+    connection a b * connection b c = connection a c := by
+  simp only [connection,hrab,hrbc,le_trans hrab hrbc,↓reduceIte]
+  exact upLabel_trans (related_forward hab hrab) (related_forward hbc hrbc)
+
+theorem connection_flat {a b c : SeedCell} (hab : Related a b) (hbc : Related b c)
+    (hac : Related a c) : connection a b * connection b c = connection a c := by
+  by_cases hrab : cellRank a ≤ cellRank b
+  · by_cases hrbc : cellRank b ≤ cellRank c
+    · exact connection_flat_sorted hab hbc hrab hrbc
+    · have hrcb := le_of_lt (lt_of_not_ge hrbc)
+      by_cases hrac : cellRank a ≤ cellRank c
+      · have H := connection_flat_sorted hac (related_symm hbc) hrac hrcb
+        rw [connection_reverse b c] at H
+        rw [← H]
+        group
+      · have hrca := le_of_lt (lt_of_not_ge hrac)
+        have H := connection_flat_sorted (related_symm hac) hab hrca hrab
+        rw [connection_reverse a c,connection_reverse b c] at H
+        calc
+          connection a b * connection b c =
+              connection a c * ((connection a c)⁻¹ * connection a b) * connection b c := by group
+          _ = connection a c * (connection b c)⁻¹ * connection b c := by rw [H]
+          _ = connection a c := by group
+  · have hrba := le_of_lt (lt_of_not_ge hrab)
+    by_cases hrac : cellRank a ≤ cellRank c
+    · have H := connection_flat_sorted (related_symm hab) hac hrba hrac
+      rw [connection_reverse a b] at H
+      rw [← H]
+      group
+    · have hrca := le_of_lt (lt_of_not_ge hrac)
+      by_cases hrbc : cellRank b ≤ cellRank c
+      · have H := connection_flat_sorted hbc (related_symm hac) hrbc hrca
+        rw [connection_reverse a c,connection_reverse a b] at H
+        calc
+          connection a b * connection b c =
+              connection a b * (connection b c * (connection a c)⁻¹) * connection a c := by group
+          _ = connection a b * (connection a b)⁻¹ * connection a c := by rw [H]
+          _ = connection a c := by group
+      · have hrcb := le_of_lt (lt_of_not_ge hrbc)
+        have H := connection_flat_sorted (related_symm hbc) (related_symm hab) hrcb hrba
+        rw [connection_reverse b c,connection_reverse a b,connection_reverse a c] at H
+        have H' := congrArg Inv.inv H
+        simpa only [mul_inv_rev,inv_inv] using H'
+
+def rankFin (a : SeedCell) : Fin 3 := ⟨cellRank a,by
+  rcases a with a | (a | a) <;> norm_num [cellRank]⟩
+
+theorem adjacent_rank_ne {a b : SeedCell} (h : seedGraph.Adj a b) : rankFin a ≠ rankFin b := by
+  intro he
+  have he' := congrArg Fin.val he
+  change cellRank a = cellRank b at he'
+  rcases h with h | h <;> have hh := properFace_rank h <;> omega
+
+theorem clique_card_le_three (s : Finset SeedCell) (hs : (s : Set SeedCell).Pairwise seedGraph.Adj) :
+    s.card ≤ 3 := by
+  have hi : Set.InjOn rankFin (s : Set SeedCell) := by
+    intro a ha b hb he
+    by_contra hn
+    exact adjacent_rank_ne (hs ha hb hn) he
+  have hc := Finset.card_le_card (Finset.subset_univ (s.image rankFin))
+  rwa [Finset.card_image_of_injOn hi,Finset.card_univ,Fintype.card_fin] at hc
+
+/-- The subdivided x-circle in the exact 103-cell order graph. -/
+def roseLoop : List SeedCell :=
+  [.inl (.inl none), .inr (.inl (.inl (false,0))),
+    .inl (.inl (some (false,0))), .inr (.inl (.inl (false,1))),
+    .inl (.inl (some (false,1))), .inr (.inl (.inl (false,2))), .inl (.inl none)]
+
+theorem roseLoop_edges : roseLoop.IsChain seedGraph.Adj := by
+  simp [roseLoop,List.isChain_cons_cons,seedGraph,properFace,edgeVertices,circleStart,circleEnd]
+
+theorem roseLoop_holonomy : (List.zipWith connection roseLoop roseLoop.tail).prod = seedXUnit := by
+  simp [roseLoop,connection,cellRank,upLabel,edgeLocal,circleLabel,circleWord,
+    circleStart,seedLetterUnit]
+
+theorem roseLoop_nontrivial : (List.zipWith connection roseLoop roseLoop.tail).prod ≠ 1 := by
+  rw [roseLoop_holonomy]
+  intro h
+  have hh := congrArg (fun q : SeedSU2 => (q : ℍ).re) h
+  norm_num [seedXUnit,seedXQuaternion] at hh
+
+end SeedConnection
+
+
+
+end EilenbergGanea
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/SeedHolonomy.lean

@@ -1,191 +1,191 @@
-import OAI.Topology.EilenbergGanea.Seed
-import OAI.Topology.EilenbergGanea.SmallHolonomy
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set Filter Topology MeasureTheory
-open scoped Quaternion ContDiff
-
-/-! Exact quaternion monodromy of the fixed presentation seed. -/
-open scoped Quaternion
-namespace EilenbergGanea
-
-abbrev SeedSU2 := unitary ℍ
-
-def seedGoldenReal : ℝ := (1 + Real.sqrt 5) / 4
-
-def seedXQuaternion : ℍ := ⟨0,1,0,0⟩
-def seedYQuaternion : ℍ :=
-  ⟨seedGoldenReal,1/2,Real.sqrt ((3 - Real.sqrt 5)/8),0⟩
-
-theorem seedGoldenReal_quadratic : 4 * seedGoldenReal ^ 2 - 2 * seedGoldenReal - 1 = 0 := by
-  have h := Real.sq_sqrt (show (0 : ℝ) ≤ 5 by norm_num)
-  dsimp [seedGoldenReal]
-  nlinarith
-
-theorem seedGoldenReal_cubic : 8 * seedGoldenReal ^ 3 - 4 * seedGoldenReal = 1 := by
-  have h := seedGoldenReal_quadratic
-  nlinarith [sq_nonneg (2 * seedGoldenReal - 1),
-    mul_eq_zero_of_left h seedGoldenReal]
-
-theorem seedGoldenReal_quartic : 16 * seedGoldenReal ^ 4 - 12 * seedGoldenReal ^ 2 + 1 = 0 := by
-  have h := seedGoldenReal_quadratic
-  have h' := seedGoldenReal_cubic
-  nlinarith [mul_eq_zero_of_left h (seedGoldenReal ^ 2),
-    congrArg (fun t : ℝ => t * seedGoldenReal) h']
-
-theorem seedX_normSq : Quaternion.normSq seedXQuaternion = 1 := by
-  rw [Quaternion.normSq_def']
-  norm_num [seedXQuaternion]
-
-theorem seedY_normSq : Quaternion.normSq seedYQuaternion = 1 := by
-  have hs := Real.sq_sqrt (show (0 : ℝ) ≤ 5 by norm_num)
-  have hn := Real.sqrt_nonneg (5 : ℝ)
-  have hle : Real.sqrt (5 : ℝ) ≤ 3 := by nlinarith
-  have hc := Real.sq_sqrt (show (0 : ℝ) ≤ (3 - Real.sqrt 5)/8 by positivity)
-  rw [Quaternion.normSq_def']
-  change seedGoldenReal ^ 2 + (1 / 2 : ℝ) ^ 2 + Real.sqrt ((3 - Real.sqrt 5)/8) ^ 2 + 0 ^ 2 = 1
-  dsimp [seedGoldenReal]
-  nlinarith
-
-def seedXUnit : SeedSU2 := ⟨seedXQuaternion,by
-  rw [Unitary.mem_iff,Quaternion.star_mul_self,Quaternion.self_mul_star,seedX_normSq]
-  simp⟩
-
-def seedYUnit : SeedSU2 := ⟨seedYQuaternion,by
-  rw [Unitary.mem_iff,Quaternion.star_mul_self,Quaternion.self_mul_star,seedY_normSq]
-  simp⟩
-
-theorem unitQuaternion_quadratic (q : ℍ) (h : Quaternion.normSq q = 1) :
-    q ^ 2 = (2 * q.re : ℝ) * q - 1 := by
-  have hh := Quaternion.star_mul_self q
-  rw [h,Quaternion.star_eq_two_re_sub,sub_mul] at hh
-  simp only [Quaternion.coe_one] at hh
-  have hh' : ((2 * q.re : ℝ) : ℍ) * q = 1 + q * q := by
-    simpa only [add_comm] using sub_eq_iff_eq_add.mp hh
-  rw [pow_two]
-  apply eq_sub_iff_add_eq.mpr
-  simpa only [add_comm] using hh'.symm
-
-theorem unitQuaternion_half_cube (q : ℍ) (h : Quaternion.normSq q = 1)
-    (hr : q.re = 1/2) : q ^ 3 = -1 := by
-  have hh := unitQuaternion_quadratic q h
-  rw [hr] at hh
-  norm_num at hh
-  calc
-    q ^ 3 = q ^ 2 * q := by rw [pow_succ]
-    _ = (q - 1) * q := by rw [hh]
-    _ = q ^ 2 - q := by noncomm_ring
-    _ = -1 := by rw [hh]; noncomm_ring
-
-theorem seedX_square : seedXQuaternion ^ 2 = -1 := by
-  rw [pow_two]
-  apply Quaternion.ext
-  · rw [Quaternion.re_mul]
-    norm_num [seedXQuaternion]
-  · rw [Quaternion.imI_mul]
-    norm_num [seedXQuaternion]
-  · rw [Quaternion.imJ_mul]
-    norm_num [seedXQuaternion]
-  · rw [Quaternion.imK_mul]
-    norm_num [seedXQuaternion]
-
-theorem seedY_fifth : seedYQuaternion ^ 5 = -1 := by
-  let q := seedYQuaternion
-  let c : ℍ := (seedGoldenReal : ℝ)
-  have hh : q ^ 2 = 2 * c * q - 1 := by
-    simpa only [q,c,seedYQuaternion,Quaternion.coe_mul, show ((2 : ℝ) : ℍ) = 2 from rfl] using unitQuaternion_quadratic seedYQuaternion seedY_normSq
-  have h3 : q ^ 3 = (4 * c ^ 2 - 1) * q - 2 * c := by
-    calc
-      q ^ 3 = q ^ 2 * q := by rw [pow_succ]
-      _ = (2 * c * q - 1) * q := by rw [hh]
-      _ = 2 * c * q ^ 2 - q := by noncomm_ring
-      _ = (4 * c ^ 2 - 1) * q - 2 * c := by rw [hh]; noncomm_ring
-  have h4 : q ^ 4 = (8 * c ^ 3 - 4 * c) * q - (4 * c ^ 2 - 1) := by
-    calc
-      q ^ 4 = q ^ 3 * q := by rw [pow_succ]
-      _ = ((4 * c ^ 2 - 1) * q - 2 * c) * q := by rw [h3]
-      _ = (4 * c ^ 2 - 1) * q ^ 2 - 2 * c * q := by noncomm_ring
-      _ = (8 * c ^ 3 - 4 * c) * q - (4 * c ^ 2 - 1) := by rw [hh]; noncomm_ring
-  have h5 : q ^ 5 = (16 * c ^ 4 - 12 * c ^ 2 + 1) * q - (8 * c ^ 3 - 4 * c) := by
-    calc
-      q ^ 5 = q ^ 4 * q := by rw [pow_succ]
-      _ = ((8 * c ^ 3 - 4 * c) * q - (4 * c ^ 2 - 1)) * q := by rw [h4]
-      _ = (8 * c ^ 3 - 4 * c) * q ^ 2 - (4 * c ^ 2 - 1) * q := by noncomm_ring
-      _ = _ := by rw [hh]; noncomm_ring
-  have hc3 : 8 * c ^ 3 - 4 * c = 1 := by
-    change (8 : ℍ) * (seedGoldenReal : ℍ) ^ 3 - 4 * (seedGoldenReal : ℍ) = 1
-    have h := congrArg (fun r : ℝ => (r : ℍ)) seedGoldenReal_cubic
-    simpa only [Quaternion.coe_sub,Quaternion.coe_mul,Quaternion.coe_pow,Quaternion.coe_one,
-      show ((8 : ℝ) : ℍ) = 8 from rfl, show ((4 : ℝ) : ℍ) = 4 from rfl] using h
-  have hc4 : 16 * c ^ 4 - 12 * c ^ 2 + 1 = 0 := by
-    change (16 : ℍ) * (seedGoldenReal : ℍ) ^ 4 - 12 * (seedGoldenReal : ℍ) ^ 2 + 1 = 0
-    have h := congrArg (fun r : ℝ => (r : ℍ)) seedGoldenReal_quartic
-    simpa only [Quaternion.coe_sub,Quaternion.coe_mul,Quaternion.coe_pow,Quaternion.coe_add,
-      Quaternion.coe_one,Quaternion.coe_zero, show ((16 : ℝ) : ℍ) = 16 from rfl,
-      show ((12 : ℝ) : ℍ) = 12 from rfl] using h
-  simpa [hc3,hc4] using h5
-
-theorem seedXYinv_cube : (seedXQuaternion * star seedYQuaternion) ^ 3 = -1 := by
-  apply unitQuaternion_half_cube
-  · rw [map_mul,Quaternion.normSq_star,seedX_normSq,seedY_normSq,mul_one]
-  · rw [Quaternion.re_mul]
-    rw [Quaternion.imI_star]
-    norm_num [seedXQuaternion,seedYQuaternion]
-
-theorem seed_unit_relations :
-    seedXUnit ^ 2 = seedYUnit ^ 5 ∧
-    seedXUnit ^ 2 = (seedXUnit * seedYUnit⁻¹) ^ 3 := by
-  constructor
-  · apply Subtype.val_injective
-    change seedXQuaternion ^ 2 = seedYQuaternion ^ 5
-    rw [seedX_square,seedY_fifth]
-  · apply Subtype.val_injective
-    change seedXQuaternion ^ 2 = (seedXQuaternion * star seedYQuaternion) ^ 3
-    rw [seedX_square,seedXYinv_cube]
-
-def seedRelator : Bool → FreeGroup Bool
-  | false => FreeGroup.of false ^ 2 * (FreeGroup.of true ^ 5)⁻¹
-  | true => FreeGroup.of false ^ 2 * ((FreeGroup.of false * (FreeGroup.of true)⁻¹) ^ 3)⁻¹
-
-theorem seedWord_relator (r : Bool) : FreeGroup.mk (seedWord r) = seedRelator r := by
-  cases r <;> decide
-
-def seedLetterUnit : Bool → SeedSU2
-  | false => seedXUnit
-  | true => seedYUnit
-
-theorem seedRelator_unit (r : Bool) : FreeGroup.lift seedLetterUnit (seedRelator r) = 1 := by
-  cases r
-  · simp only [seedRelator,map_mul,map_inv,map_pow,FreeGroup.lift_apply_of,seedLetterUnit]
-    rw [seed_unit_relations.1,mul_inv_cancel]
-  · simp only [seedRelator,map_mul,map_inv,map_pow,FreeGroup.lift_apply_of,seedLetterUnit]
-    rw [seed_unit_relations.2,mul_inv_cancel]
-
-abbrev SeedPresentationGroup := PresentedGroup (Set.range seedRelator)
-
-def seedMonodromy : SeedPresentationGroup →* SeedSU2 :=
-  PresentedGroup.toGroup (f := seedLetterUnit) (by
-    rintro _ ⟨r,rfl⟩
-    exact seedRelator_unit r)
-
-theorem seedMonodromy_nontrivial : seedMonodromy (PresentedGroup.of false) ≠ 1 := by
-  intro h
-  have hh := congrArg (fun q : SeedSU2 => (q : ℍ).re) h
-  simp [seedMonodromy,seedLetterUnit,seedXUnit,seedXQuaternion] at hh
-
-/-- The source exponent-sum boundary, with rows indexed by its two relators. -/
-def seedExponentMatrix : Matrix (Fin 2) (Fin 2) ℤ := !![2,-5;-1,3]
-
-theorem seedExponentMatrix_det : seedExponentMatrix.det = 1 := by
-  norm_num [seedExponentMatrix,Matrix.det_fin_two]
-
-end EilenbergGanea
-
-
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.Seed
+import OAI.Topology.EilenbergGanea.SmallHolonomy
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set Filter Topology MeasureTheory
+open scoped Quaternion ContDiff
+
+/-! Exact quaternion monodromy of the fixed presentation seed. -/
+open scoped Quaternion
+namespace EilenbergGanea
+
+abbrev SeedSU2 := unitary ℍ
+
+def seedGoldenReal : ℝ := (1 + Real.sqrt 5) / 4
+
+def seedXQuaternion : ℍ := ⟨0,1,0,0⟩
+def seedYQuaternion : ℍ :=
+  ⟨seedGoldenReal,1/2,Real.sqrt ((3 - Real.sqrt 5)/8),0⟩
+
+theorem seedGoldenReal_quadratic : 4 * seedGoldenReal ^ 2 - 2 * seedGoldenReal - 1 = 0 := by
+  have h := Real.sq_sqrt (show (0 : ℝ) ≤ 5 by norm_num)
+  dsimp [seedGoldenReal]
+  nlinarith
+
+theorem seedGoldenReal_cubic : 8 * seedGoldenReal ^ 3 - 4 * seedGoldenReal = 1 := by
+  have h := seedGoldenReal_quadratic
+  nlinarith [sq_nonneg (2 * seedGoldenReal - 1),
+    mul_eq_zero_of_left h seedGoldenReal]
+
+theorem seedGoldenReal_quartic : 16 * seedGoldenReal ^ 4 - 12 * seedGoldenReal ^ 2 + 1 = 0 := by
+  have h := seedGoldenReal_quadratic
+  have h' := seedGoldenReal_cubic
+  nlinarith [mul_eq_zero_of_left h (seedGoldenReal ^ 2),
+    congrArg (fun t : ℝ => t * seedGoldenReal) h']
+
+theorem seedX_normSq : Quaternion.normSq seedXQuaternion = 1 := by
+  rw [Quaternion.normSq_def']
+  norm_num [seedXQuaternion]
+
+theorem seedY_normSq : Quaternion.normSq seedYQuaternion = 1 := by
+  have hs := Real.sq_sqrt (show (0 : ℝ) ≤ 5 by norm_num)
+  have hn := Real.sqrt_nonneg (5 : ℝ)
+  have hle : Real.sqrt (5 : ℝ) ≤ 3 := by nlinarith
+  have hc := Real.sq_sqrt (show (0 : ℝ) ≤ (3 - Real.sqrt 5)/8 by positivity)
+  rw [Quaternion.normSq_def']
+  change seedGoldenReal ^ 2 + (1 / 2 : ℝ) ^ 2 + Real.sqrt ((3 - Real.sqrt 5)/8) ^ 2 + 0 ^ 2 = 1
+  dsimp [seedGoldenReal]
+  nlinarith
+
+def seedXUnit : SeedSU2 := ⟨seedXQuaternion,by
+  rw [Unitary.mem_iff,Quaternion.star_mul_self,Quaternion.self_mul_star,seedX_normSq]
+  simp⟩
+
+def seedYUnit : SeedSU2 := ⟨seedYQuaternion,by
+  rw [Unitary.mem_iff,Quaternion.star_mul_self,Quaternion.self_mul_star,seedY_normSq]
+  simp⟩
+
+theorem unitQuaternion_quadratic (q : ℍ) (h : Quaternion.normSq q = 1) :
+    q ^ 2 = (2 * q.re : ℝ) * q - 1 := by
+  have hh := Quaternion.star_mul_self q
+  rw [h,Quaternion.star_eq_two_re_sub,sub_mul] at hh
+  simp only [Quaternion.coe_one] at hh
+  have hh' : ((2 * q.re : ℝ) : ℍ) * q = 1 + q * q := by
+    simpa only [add_comm] using sub_eq_iff_eq_add.mp hh
+  rw [pow_two]
+  apply eq_sub_iff_add_eq.mpr
+  simpa only [add_comm] using hh'.symm
+
+theorem unitQuaternion_half_cube (q : ℍ) (h : Quaternion.normSq q = 1)
+    (hr : q.re = 1/2) : q ^ 3 = -1 := by
+  have hh := unitQuaternion_quadratic q h
+  rw [hr] at hh
+  norm_num at hh
+  calc
+    q ^ 3 = q ^ 2 * q := by rw [pow_succ]
+    _ = (q - 1) * q := by rw [hh]
+    _ = q ^ 2 - q := by noncomm_ring
+    _ = -1 := by rw [hh]; noncomm_ring
+
+theorem seedX_square : seedXQuaternion ^ 2 = -1 := by
+  rw [pow_two]
+  apply Quaternion.ext
+  · rw [Quaternion.re_mul]
+    norm_num [seedXQuaternion]
+  · rw [Quaternion.imI_mul]
+    norm_num [seedXQuaternion]
+  · rw [Quaternion.imJ_mul]
+    norm_num [seedXQuaternion]
+  · rw [Quaternion.imK_mul]
+    norm_num [seedXQuaternion]
+
+theorem seedY_fifth : seedYQuaternion ^ 5 = -1 := by
+  let q := seedYQuaternion
+  let c : ℍ := (seedGoldenReal : ℝ)
+  have hh : q ^ 2 = 2 * c * q - 1 := by
+    simpa only [q,c,seedYQuaternion,Quaternion.coe_mul, show ((2 : ℝ) : ℍ) = 2 from rfl] using unitQuaternion_quadratic seedYQuaternion seedY_normSq
+  have h3 : q ^ 3 = (4 * c ^ 2 - 1) * q - 2 * c := by
+    calc
+      q ^ 3 = q ^ 2 * q := by rw [pow_succ]
+      _ = (2 * c * q - 1) * q := by rw [hh]
+      _ = 2 * c * q ^ 2 - q := by noncomm_ring
+      _ = (4 * c ^ 2 - 1) * q - 2 * c := by rw [hh]; noncomm_ring
+  have h4 : q ^ 4 = (8 * c ^ 3 - 4 * c) * q - (4 * c ^ 2 - 1) := by
+    calc
+      q ^ 4 = q ^ 3 * q := by rw [pow_succ]
+      _ = ((4 * c ^ 2 - 1) * q - 2 * c) * q := by rw [h3]
+      _ = (4 * c ^ 2 - 1) * q ^ 2 - 2 * c * q := by noncomm_ring
+      _ = (8 * c ^ 3 - 4 * c) * q - (4 * c ^ 2 - 1) := by rw [hh]; noncomm_ring
+  have h5 : q ^ 5 = (16 * c ^ 4 - 12 * c ^ 2 + 1) * q - (8 * c ^ 3 - 4 * c) := by
+    calc
+      q ^ 5 = q ^ 4 * q := by rw [pow_succ]
+      _ = ((8 * c ^ 3 - 4 * c) * q - (4 * c ^ 2 - 1)) * q := by rw [h4]
+      _ = (8 * c ^ 3 - 4 * c) * q ^ 2 - (4 * c ^ 2 - 1) * q := by noncomm_ring
+      _ = _ := by rw [hh]; noncomm_ring
+  have hc3 : 8 * c ^ 3 - 4 * c = 1 := by
+    change (8 : ℍ) * (seedGoldenReal : ℍ) ^ 3 - 4 * (seedGoldenReal : ℍ) = 1
+    have h := congrArg (fun r : ℝ => (r : ℍ)) seedGoldenReal_cubic
+    simpa only [Quaternion.coe_sub,Quaternion.coe_mul,Quaternion.coe_pow,Quaternion.coe_one,
+      show ((8 : ℝ) : ℍ) = 8 from rfl, show ((4 : ℝ) : ℍ) = 4 from rfl] using h
+  have hc4 : 16 * c ^ 4 - 12 * c ^ 2 + 1 = 0 := by
+    change (16 : ℍ) * (seedGoldenReal : ℍ) ^ 4 - 12 * (seedGoldenReal : ℍ) ^ 2 + 1 = 0
+    have h := congrArg (fun r : ℝ => (r : ℍ)) seedGoldenReal_quartic
+    simpa only [Quaternion.coe_sub,Quaternion.coe_mul,Quaternion.coe_pow,Quaternion.coe_add,
+      Quaternion.coe_one,Quaternion.coe_zero, show ((16 : ℝ) : ℍ) = 16 from rfl,
+      show ((12 : ℝ) : ℍ) = 12 from rfl] using h
+  simpa [hc3,hc4] using h5
+
+theorem seedXYinv_cube : (seedXQuaternion * star seedYQuaternion) ^ 3 = -1 := by
+  apply unitQuaternion_half_cube
+  · rw [map_mul,Quaternion.normSq_star,seedX_normSq,seedY_normSq,mul_one]
+  · rw [Quaternion.re_mul]
+    rw [Quaternion.imI_star]
+    norm_num [seedXQuaternion,seedYQuaternion]
+
+theorem seed_unit_relations :
+    seedXUnit ^ 2 = seedYUnit ^ 5 ∧
+    seedXUnit ^ 2 = (seedXUnit * seedYUnit⁻¹) ^ 3 := by
+  constructor
+  · apply Subtype.val_injective
+    change seedXQuaternion ^ 2 = seedYQuaternion ^ 5
+    rw [seedX_square,seedY_fifth]
+  · apply Subtype.val_injective
+    change seedXQuaternion ^ 2 = (seedXQuaternion * star seedYQuaternion) ^ 3
+    rw [seedX_square,seedXYinv_cube]
+
+def seedRelator : Bool → FreeGroup Bool
+  | false => FreeGroup.of false ^ 2 * (FreeGroup.of true ^ 5)⁻¹
+  | true => FreeGroup.of false ^ 2 * ((FreeGroup.of false * (FreeGroup.of true)⁻¹) ^ 3)⁻¹
+
+theorem seedWord_relator (r : Bool) : FreeGroup.mk (seedWord r) = seedRelator r := by
+  cases r <;> decide
+
+def seedLetterUnit : Bool → SeedSU2
+  | false => seedXUnit
+  | true => seedYUnit
+
+theorem seedRelator_unit (r : Bool) : FreeGroup.lift seedLetterUnit (seedRelator r) = 1 := by
+  cases r
+  · simp only [seedRelator,map_mul,map_inv,map_pow,FreeGroup.lift_apply_of,seedLetterUnit]
+    rw [seed_unit_relations.1,mul_inv_cancel]
+  · simp only [seedRelator,map_mul,map_inv,map_pow,FreeGroup.lift_apply_of,seedLetterUnit]
+    rw [seed_unit_relations.2,mul_inv_cancel]
+
+abbrev SeedPresentationGroup := PresentedGroup (Set.range seedRelator)
+
+def seedMonodromy : SeedPresentationGroup →* SeedSU2 :=
+  PresentedGroup.toGroup (f := seedLetterUnit) (by
+    rintro _ ⟨r,rfl⟩
+    exact seedRelator_unit r)
+
+theorem seedMonodromy_nontrivial : seedMonodromy (PresentedGroup.of false) ≠ 1 := by
+  intro h
+  have hh := congrArg (fun q : SeedSU2 => (q : ℍ).re) h
+  simp [seedMonodromy,seedLetterUnit,seedXUnit,seedXQuaternion] at hh
+
+/-- The source exponent-sum boundary, with rows indexed by its two relators. -/
+def seedExponentMatrix : Matrix (Fin 2) (Fin 2) ℤ := !![2,-5;-1,3]
+
+theorem seedExponentMatrix_det : seedExponentMatrix.det = 1 := by
+  norm_num [seedExponentMatrix,Matrix.det_fin_two]
+
+end EilenbergGanea
+
+
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/SmallHolonomy.lean

@@ -1,744 +1,744 @@
-import OAI.Topology.EilenbergGanea.WordEquations
-import OAI.GroupTheory.RightAngledArtin.ResidualFiniteness
-import Mathlib.Topology.Compactness.Compact
-import Mathlib.LinearAlgebra.Finsupp.LinearCombination
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set Filter Topology MeasureTheory
-open scoped Quaternion ContDiff
-
-namespace EilenbergGanea
-open scoped Quaternion ContDiff
-/-- Squared chord length on the unit-quaternion sphere. -/
-theorem su2_distance_square (a : SU2) :
-    ‖(a : ℍ) - 1‖ ^ 2 = 2 - 2 * (a : ℍ).re := by
-  have hu := (quaternion_mem_unitary_iff _).mp a.property
-  have hs : Quaternion.normSq ((a : ℍ) - 1) = Quaternion.normSq (a : ℍ) + 1 - 2 * (a : ℍ).re := by
-    simp only [Quaternion.normSq_def', Quaternion.re_sub, Quaternion.imI_sub,
-      Quaternion.imJ_sub, Quaternion.imK_sub, Quaternion.re_one,
-      Quaternion.imI_one, Quaternion.imJ_one, Quaternion.imK_one]
-    ring
-  simpa only [Quaternion.normSq_eq_norm_mul_self, hu, one_mul, sq, one_add_one_eq_two] using hs
-
-def arcBlend (a : SU2) (t : ℝ) : ℍ :=
-  (1 - Real.smoothTransition t) • (1 : ℍ) + Real.smoothTransition t • (a : ℍ)
-
-theorem arcBlend_norm_le (a : SU2) (t : ℝ) : ‖arcBlend a t‖ ≤ 1 := by
-  have h0 := Real.smoothTransition.nonneg t
-  have h1 := Real.smoothTransition.le_one t
-  have hn := (quaternion_mem_unitary_iff _).mp a.property
-  calc
-    _ ≤ ‖(1 - Real.smoothTransition t) • (1 : ℍ)‖ +
-        ‖Real.smoothTransition t • (a : ℍ)‖ := norm_add_le _ _
-    _ = 1 := by rw [norm_smul, norm_smul, norm_one, hn, Real.norm_eq_abs,
-        Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr h1), abs_of_nonneg h0]; ring
-
-theorem small_unit_re (a : SU2) (ha : ‖(a : ℍ) - 1‖ < 1) :
-    0 < (a : ℍ).re ∧ (a : ℍ).re ≤ 1 := by
-  have h := su2_distance_square a
-  have hn := norm_nonneg ((a : ℍ) - 1)
-  constructor <;> nlinarith [sq_nonneg ‖(a : ℍ) - 1‖]
-
-theorem arcBlend_re_le (a : SU2) (ha : ‖(a : ℍ) - 1‖ < 1) (t : ℝ) :
-    (a : ℍ).re ≤ (arcBlend a t).re := by
-  have h := (small_unit_re a ha).2
-  have h0 := Real.smoothTransition.nonneg t
-  have h1 := Real.smoothTransition.le_one t
-  simp only [arcBlend, Quaternion.re_add, Quaternion.re_smul, Quaternion.re_one, smul_eq_mul]
-  nlinarith
-
-theorem arcBlend_ne_zero (a : SU2) (ha : ‖(a : ℍ) - 1‖ < 1) (t : ℝ) :
-    arcBlend a t ≠ 0 := by
-  have h := (small_unit_re a ha).1.trans_le (arcBlend_re_le a ha t)
-  intro he
-  simp only [he, Quaternion.re_zero, lt_self_iff_false] at h
-
-/-- A globally smooth path that remains in the closed chord ball of its
-endpoint, avoiding any branch choice of a quaternion logarithm. -/
-def smallQuaternionArc (a : SU2) (t : ℝ) : SU2 := normalizeQuaternion (arcBlend a t)
-
-@[simp] theorem smallQuaternionArc_zero (a : SU2) : smallQuaternionArc a 0 = 1 := by
-  simpa [smallQuaternionArc, arcBlend] using (normalizeQuaternion_unitary (1 : SU2))
-
-@[simp] theorem smallQuaternionArc_one (a : SU2) : smallQuaternionArc a 1 = a := by
-  simp [smallQuaternionArc, arcBlend]
-
-theorem smallQuaternionArc_smooth (a : SU2) (ha : ‖(a : ℍ) - 1‖ < 1) :
-    ContDiff ℝ ∞ (fun t => (smallQuaternionArc a t : ℍ)) := by
-  apply contDiff_iff_contDiffAt.mpr
-  intro t
-  exact (normalizeQuaternion_coe_contDiffAt (arcBlend_ne_zero a ha t)).comp t
-    (((contDiff_const.sub Real.smoothTransition.contDiff).smul contDiff_const).add
-      (Real.smoothTransition.contDiff.smul contDiff_const)).contDiffAt
-
-theorem smallQuaternionArc_distance (a : SU2) (ha : ‖(a : ℍ) - 1‖ < 1) (t : ℝ) :
-    ‖(smallQuaternionArc a t : ℍ) - 1‖ ≤ ‖(a : ℍ) - 1‖ := by
-  have hp : 0 < ‖arcBlend a t‖ := norm_pos_iff.mpr (arcBlend_ne_zero a ha t)
-  have hre : (a : ℍ).re ≤ (smallQuaternionArc a t : ℍ).re := by
-    rw [smallQuaternionArc, normalizeQuaternion_coe_of_ne_zero (arcBlend_ne_zero a ha t)]
-    simp only [Quaternion.re_smul, smul_eq_mul]
-    rw [mul_comm, ← div_eq_mul_inv, le_div_iff₀ hp]
-    exact (mul_le_of_le_one_right (small_unit_re a ha).1.le (arcBlend_norm_le a t)).trans
-      (arcBlend_re_le a ha t)
-  have h0 := su2_distance_square a
-  have h1 := su2_distance_square (smallQuaternionArc a t)
-  nlinarith [norm_nonneg ((a : ℍ)-1), norm_nonneg ((smallQuaternionArc a t : ℍ)-1)]
-
-end EilenbergGanea
-
-namespace EilenbergGanea
-open scoped Quaternion ContDiff
-open Set
-section SelectedEquations
-variable {E : Type*} {m d : ℕ}
-
-/-- The selected unknowns remain separate from coefficients, even when two
-letters have the same image in the group quotient. -/
-def selectedAlphabet (σ : Fin m ↪ E) (e : E) : E ⊕ Fin m := by
-  classical
-  exact if h : ∃ i, σ i = e then .inr h.choose else .inl e
-
-@[simp] theorem selectedAlphabet_selected (σ : Fin m ↪ E) (i : Fin m) :
-    selectedAlphabet σ (σ i) = .inr i := by
-  classical
-  have h : ∃ j, σ j = σ i := ⟨i,rfl⟩
-  simp only [selectedAlphabet, dite_eq_left h]
-  exact congrArg Sum.inr (σ.injective h.choose_spec)
-
- theorem selectedAlphabet_other (σ : Fin m ↪ E) (e : E) (he : ∀ i, σ i ≠ e) :
-    selectedAlphabet σ e = .inl e := by
-  classical
-  simp [selectedAlphabet, he]
-
-def selectedWord (σ : Fin m ↪ E) : FreeGroup E →* FreeGroup (E ⊕ Fin m) :=
-  FreeGroup.map (selectedAlphabet σ)
-
-def selectedLabel (σ : Fin m ↪ E) (a : E → SU2) (x : Fin m → SU2) (e : E) : SU2 :=
-  Sum.elim a x (selectedAlphabet σ e)
-
-@[simp] theorem selectedLabel_selected (σ : Fin m ↪ E) (a : E → SU2)
-    (x : Fin m → SU2) (i : Fin m) : selectedLabel σ a x (σ i) = x i := by
-  simp [selectedLabel]
-
- theorem selectedLabel_other (σ : Fin m ↪ E) (a : E → SU2) (x : Fin m → SU2)
-    (e : E) (he : ∀ i, σ i ≠ e) : selectedLabel σ a x e = a e := by
-  rw [selectedLabel, selectedAlphabet_other σ e he]
-  rfl
-
- theorem selectedWord_eval (σ : Fin m ↪ E) (a : E → SU2) (x : Fin m → SU2)
-    (w : FreeGroup E) :
-    FreeGroup.lift (Sum.elim a x) (selectedWord σ w) =
-      FreeGroup.lift (selectedLabel σ a x) w := by
-  have he : (FreeGroup.lift (Sum.elim a x)).comp (selectedWord σ) =
-      FreeGroup.lift (selectedLabel σ a x) := by
-    apply FreeGroup.ext_hom
-    intro e
-    simp [selectedWord, selectedLabel]
-  exact congrArg (fun f : FreeGroup E →* SU2 => f w) he
-
- theorem selectedWord_exponents [DecidableEq E] (σ : Fin m ↪ E) (w : FreeGroup E) (j : Fin m) :
-    wordExponents (eraseCoefficients (selectedWord σ w)) j = wordExponents w (σ j) := by
-  classical
-  have he : (FreeGroup.lift (fun i : Fin m => Multiplicative.ofAdd (if j = i then (1 : ℤ) else 0))).comp
-      (eraseCoefficients.comp (selectedWord σ)) =
-      FreeGroup.lift (fun e : E => Multiplicative.ofAdd (if σ j = e then (1 : ℤ) else 0)) := by
-    apply FreeGroup.ext_hom
-    intro e
-    by_cases h : ∃ i, σ i = e
-    · obtain ⟨i,rfl⟩ := h
-      simp [selectedWord, eraseCoefficients, σ.injective.eq_iff]
-    · have h' : ∀ i, σ i ≠ e := by simpa only [not_exists] using h
-      simp [selectedWord, eraseCoefficients, selectedAlphabet_other σ e h', h' j]
-  exact congrArg (fun f : FreeGroup E →* Multiplicative ℤ => (f w).toAdd) he
-
-/-- Solve a finite, full-rank system while keeping every unselected label fixed.
-The distinguished sublevel condition is the actual isolated root condition,
-not a nonsingularity assumption on the roots. -/
-theorem selected_word_sublevel_solvable [DecidableEq E]
-    (σ : Fin m ↪ E) (w : Fin m → FreeGroup E) (ρ : Fin d → FreeGroup E)
-    (a : ℝ → E → SU2) (ha : ∀ e, ContDiff ℝ ∞ (fun t => (a t e : ℍ)))
-    (ha0 : ∀ e, a 0 e = 1)
-    (hB : Matrix.det (fun i j => wordExponents (w i) (σ j)) ≠ 0)
-    (δ : ℝ) (hδ : 0 < δ)
-    (hiso : ∀ t ∈ Icc (0 : ℝ) 1, ∀ x : Fin m → SU2,
-      (∀ i, FreeGroup.lift (selectedLabel σ (a t) x) (w i) = 1) →
-      ‖(fun i => (FreeGroup.lift (selectedLabel σ (a t) x) (ρ i) : ℍ)) - 1‖ ≠ δ) :
-    ∃ b : E → SU2,
-      (∀ e, (∀ i, σ i ≠ e) → b e = a 1 e) ∧
-      (∀ i, FreeGroup.lift b (w i) = 1) ∧
-      ‖(fun i => (FreeGroup.lift b (ρ i) : ℍ)) - 1‖ < δ := by
-  have hb : (exponentMatrix (fun i => eraseCoefficients (selectedWord σ (w i)))).det ≠ 0 := by
-    change (Matrix.det (fun i j => wordExponents (eraseCoefficients (selectedWord σ (w i))) j)) ≠ 0
-    simpa only [selectedWord_exponents] using hB
-  obtain ⟨x,hx,hρ⟩ := coefficient_word_sublevel_solvable
-    (fun i => selectedWord σ (w i)) (fun i => selectedWord σ (ρ i)) a ha ha0 hb δ hδ
-    (fun t ht x hx => by
-      simp only [coefficientWordMap, selectedWord_eval]
-      apply hiso t ht x
-      intro i
-      have hi := congrFun hx i
-      simpa only [coefficientWordMap, selectedWord_eval, Pi.one_apply] using hi)
-  refine ⟨selectedLabel σ (a 1) x, fun e he => selectedLabel_other σ _ x e he, ?_, ?_⟩
-  · intro i
-    have hi := congrFun hx i
-    simpa only [coefficientWordMap, selectedWord_eval, Pi.one_apply] using hi
-  · simpa only [coefficientWordMap, selectedWord_eval] using hρ
-
-end SelectedEquations
-end EilenbergGanea
-
-namespace EilenbergGanea
-
-variable {Γ S H : Type*} [Group Γ] [Group H]
-
-def labelLift (v : S → Γ) (a : Γ → S → H) :
-    FreeGroup S →* ((Γ → H) ⋊[functionShift Γ H] Γ) :=
-  FreeGroup.lift (fun s => ⟨fun g => a g s, v s⟩)
-
-def wordValue (v : S → Γ) : FreeGroup S →* Γ := FreeGroup.lift v
-
-def pathProduct (v : S → Γ) (a : Γ → S → H) (w : FreeGroup S) (g : Γ) : H :=
-  (labelLift v a w).left g
-
- theorem labelLift_right (v : S → Γ) (a : Γ → S → H) (w : FreeGroup S) :
-    (labelLift v a w).right = wordValue v w := by
-  have he : (SemidirectProduct.rightHom : ((Γ → H) ⋊[functionShift Γ H] Γ) →* Γ).comp
-      (labelLift v a) = wordValue v := by
-    apply FreeGroup.ext_hom
-    intro s
-    simp [labelLift, wordValue]
-  exact congrArg (fun f : FreeGroup S →* Γ => f w) he
-
-@[simp] theorem pathProduct_one (v : S → Γ) (a : Γ → S → H) (g : Γ) :
-    pathProduct v a 1 g = 1 := by simp [pathProduct]
-@[simp] theorem pathProduct_of (v : S → Γ) (a : Γ → S → H) (s : S) (g : Γ) :
-    pathProduct v a (FreeGroup.of s) g = a g s := by simp [pathProduct, labelLift]
- theorem pathProduct_mul (v : S → Γ) (a : Γ → S → H) (w z : FreeGroup S) (g : Γ) :
-    pathProduct v a (w*z) g = pathProduct v a w g * pathProduct v a z (g * wordValue v w) := by
-  simp only [pathProduct, map_mul, SemidirectProduct.mul_left, Pi.mul_apply, labelLift_right]
-  rfl
- theorem pathProduct_inv (v : S → Γ) (a : Γ → S → H) (w : FreeGroup S) (g : Γ) :
-    pathProduct v a w⁻¹ g = (pathProduct v a w (g * (wordValue v w)⁻¹))⁻¹ := by
-  simp only [pathProduct, map_inv, SemidirectProduct.inv_left, labelLift_right]
-  rfl
-
- theorem pathProduct_conjugate_loop (v : S → Γ) (a : Γ → S → H)
-    (u r : FreeGroup S) (hr : wordValue v r = 1) (g : Γ) :
-    pathProduct v a (u*r*u⁻¹) g = pathProduct v a u g *
-      pathProduct v a r (g * wordValue v u) * (pathProduct v a u g)⁻¹ := by
-  rw [pathProduct_mul, pathProduct_mul, pathProduct_inv, map_mul, hr, mul_one]
-  simp only [mul_inv_cancel_right]
-
- theorem pathProduct_continuous [TopologicalSpace H] [IsTopologicalGroup H]
-    (v : S → Γ) (w : FreeGroup S) (g : Γ) :
-    Continuous (fun a : Γ → S → H => pathProduct v a w g) := by
-  induction w using FreeGroup.induction_on generalizing g with
-  | one => simpa using (continuous_const : Continuous (fun _ : Γ → S → H => (1 : H)))
-  | of s =>
-      simp only [pathProduct_of]
-      exact continuous_apply_apply g s
-  | inv_of s =>
-      simp only [pathProduct_inv, pathProduct_of]
-      exact (continuous_apply_apply (g * (wordValue v (FreeGroup.of s))⁻¹) s).inv
-  | mul w z hw hz =>
-      simp only [pathProduct_mul]
-      exact (hw g).mul (hz (g * wordValue v w))
-
-/-- The set of vertices inspected by a word is finite, independently of the
-label values. No countability hypothesis on the alphabet is needed. -/
-theorem pathProduct_finite_locations (v : S → Γ) (w : FreeGroup S) (g : Γ) :
-    ∃ P : Finset Γ, ∀ a b : Γ → S → H,
-      (∀ p ∈ P, ∀ s, a p s = b p s) → pathProduct v a w g = pathProduct v b w g := by
-  classical
-  induction w using FreeGroup.induction_on generalizing g with
-  | one => exact ⟨∅, fun _ _ _ => by simp⟩
-  | of s => exact ⟨{g}, fun a b hab => by simpa using hab g (by simp) s⟩
-  | inv_of s =>
-      exact ⟨{g * (wordValue v (FreeGroup.of s))⁻¹}, fun a b hab => by
-        rw [pathProduct_inv, pathProduct_inv, pathProduct_of, pathProduct_of, hab _ (by simp)]⟩
-  | mul w z hw hz =>
-      obtain ⟨P, hP⟩ := hw g
-      obtain ⟨Q, hQ⟩ := hz (g * wordValue v w)
-      refine ⟨P ∪ Q, fun a b hab => ?_⟩
-      rw [pathProduct_mul, pathProduct_mul, hP a b (fun p hp => hab p (Finset.mem_union_left Q hp)),
-        hQ a b (fun p hp => hab p (Finset.mem_union_right P hp))]
-
-/-- Evaluation commutes with pullback of a labeling along a group quotient,
-including when different alphabet letters have the same image. -/
-theorem pathProduct_pullback {D : Type*} [Group D] (q : Γ →* D)
-    (v : S → Γ) (a : D → S → H) (w : FreeGroup S) (g : Γ) :
-    pathProduct v (fun p s => a (q p) s) w g = pathProduct (fun s => q (v s)) a w (q g) := by
-  have hv : ∀ w : FreeGroup S, wordValue (fun s => q (v s)) w = q (wordValue v w) := by
-    have he : wordValue (fun s => q (v s)) = q.comp (wordValue v) := by
-      apply FreeGroup.ext_hom
-      intro s
-      simp [wordValue]
-    intro w
-    exact congrArg (fun f : FreeGroup S →* D => f w) he
-  induction w using FreeGroup.induction_on generalizing g with
-  | one => simp
-  | of s => simp
-  | inv_of s => simp only [pathProduct_inv, pathProduct_of, hv, map_inv, map_mul]
-  | mul w z hw hz => simp only [pathProduct_mul, hv, map_mul, hw, hz]
-
-
-/-- Once every final relator has trivial transport at every vertex, normal
-closure gives trivial transport on every null word, not just on a finite
-collection of test loops. -/
- theorem pathProduct_trivial_of_relators (v : S → Γ) (a : Γ → S → H)
-    (R : Set (FreeGroup S)) (hR : R ⊆ (wordValue v).ker)
-    (hp : (wordValue v).ker = Subgroup.normalClosure R)
-    (ha : ∀ r ∈ R, ∀ g, pathProduct v a r g = 1)
-    (w : FreeGroup S) (hw : wordValue v w = 1) (g : Γ) : pathProduct v a w g = 1 := by
-  have hrel : R ⊆ (labelLift v a).ker := by
-    intro r hr
-    change labelLift v a r = 1
-    apply SemidirectProduct.ext
-    · funext g
-      exact ha r hr g
-    · rw [labelLift_right]
-      exact hR hr
-  have hle : Subgroup.normalClosure R ≤ (labelLift v a).ker :=
-    Subgroup.normalClosure_le_normal hrel
-  have he : labelLift v a w = 1 := hle (by rw [← hp]; exact hw)
-  change (labelLift v a w).left g = 1
-  rw [he]
-  rfl
-
- def pathChain (v : S → Γ) (w : FreeGroup S) (g : Γ) : (Γ × S) →₀ ℤ :=
-  (pathProduct v (fun p s => Multiplicative.ofAdd (Finsupp.single (p,s) 1)) w g).toAdd
-
-@[simp] theorem pathChain_one (v : S → Γ) (g : Γ) : pathChain v 1 g = 0 := by
-  simp [pathChain]
-@[simp] theorem pathChain_of (v : S → Γ) (s : S) (g : Γ) :
-    pathChain v (FreeGroup.of s) g = Finsupp.single (g,s) 1 := by simp [pathChain]
- theorem pathChain_mul (v : S → Γ) (w z : FreeGroup S) (g : Γ) :
-    pathChain v (w*z) g = pathChain v w g + pathChain v z (g * wordValue v w) := by
-  simp only [pathChain, pathProduct_mul, toAdd_mul]
- theorem pathChain_inv (v : S → Γ) (w : FreeGroup S) (g : Γ) :
-    pathChain v w⁻¹ g = -pathChain v w (g * (wordValue v w)⁻¹) := by
-  simp only [pathChain, pathProduct_inv, toAdd_inv]
- theorem pathChain_conjugate (v : S → Γ) (u r : FreeGroup S)
-    (hr : wordValue v r = 1) (g : Γ) :
-    pathChain v (u*r*u⁻¹) g = pathChain v r (g * wordValue v u) := by
-  simp only [pathChain, pathProduct_conjugate_loop v _ u r hr g,
-    toAdd_mul, toAdd_inv]
-  abel
-
-variable {R : Type*}
-abbrev NormalFactor (S R : Type*) := FreeGroup S × R × Bool
-
- def normalFactorWord (r : R → FreeGroup S) (f : NormalFactor S R) : FreeGroup S :=
-  f.1 * (if f.2.2 then r f.2.1 else (r f.2.1)⁻¹) * f.1⁻¹
-
- def normalFactorCount (v : S → Γ) (f : NormalFactor S R) (g : Γ) : (Γ × R) →₀ ℤ :=
-  Finsupp.single (g * wordValue v f.1, f.2.1) (if f.2.2 then 1 else -1)
-
- def relatorChainMap (v : S → Γ) (r : R → FreeGroup S) :
-    ((Γ × R) →₀ ℤ) →ₗ[ℤ] ((Γ × S) →₀ ℤ) :=
-  Finsupp.linearCombination ℤ (fun i => pathChain v (r i.2) i.1)
-
- theorem normalFactor_value (v : S → Γ) (r : R → FreeGroup S)
-    (hr : ∀ i, wordValue v (r i) = 1) (f : NormalFactor S R) :
-    wordValue v (normalFactorWord r f) = 1 := by
-  dsimp only [normalFactorWord]
-  split_ifs <;> simp [hr]
-
- theorem pathChain_normalFactor (v : S → Γ) (r : R → FreeGroup S)
-    (hr : ∀ i, wordValue v (r i) = 1) (f : NormalFactor S R) (g : Γ) :
-    pathChain v (normalFactorWord r f) g = relatorChainMap v r (normalFactorCount v f g) := by
-  simp only [normalFactorWord, normalFactorCount, relatorChainMap, Finsupp.linearCombination_single]
-  split_ifs with h
-  · rw [pathChain_conjugate v _ _ (hr _)]; simp only [one_smul]
-  · rw [pathChain_conjugate v _ _ (by simp [hr]), pathChain_inv, hr]
-    simp only [inv_one, mul_one, neg_one_smul]
-
- theorem pathChain_normalProduct (v : S → Γ) (r : R → FreeGroup S)
-    (hr : ∀ i, wordValue v (r i) = 1) (l : List (NormalFactor S R)) (g : Γ) :
-    pathChain v (l.map (normalFactorWord r)).prod g =
-      relatorChainMap v r (l.map (fun f => normalFactorCount v f g)).sum := by
-  induction l with
-  | nil => simp
-  | cons f l ih =>
-      simp only [List.map_cons, List.prod_cons, List.sum_cons, map_add, pathChain_mul,
-        normalFactor_value v r hr f, mul_one, pathChain_normalFactor v r hr f g, ih]
-
-/-- The coefficient identity is an equality in the full free abelian group of
-translated indexed relators. Injectivity is sufficient; no finiteness of the
-relator or generator types is assumed. -/
- theorem exact_translated_relator_counts (v : S → Γ) (r : R → FreeGroup S)
-    (hr : ∀ i, wordValue v (r i) = 1) (hinj : Function.Injective (relatorChainMap v r))
-    (z : FreeGroup S) (ρ : R) (hchain : pathChain v z 1 = pathChain v (r ρ) 1)
-    (l : List (NormalFactor S R)) (he : z = (l.map (normalFactorWord r)).prod) :
-    (l.map (fun f => normalFactorCount v f 1)).sum = Finsupp.single (1,ρ) 1 := by
-  apply hinj
-  rw [← pathChain_normalProduct v r hr, ← he, hchain]
-  simp [relatorChainMap]
-
-end EilenbergGanea
-
-namespace EilenbergGanea
-variable {Γ S H : Type*} [Group Γ] [Group H]
-
-
-section CompactLabeling
-variable {T I : Type*}
-
- def mergeLabels (a : Γ → S → H) (b : Γ → T → H) : Γ → S ⊕ T → H :=
-  fun g => Sum.elim (a g) (b g)
-
- omit [Group Γ] [Group H] in
- theorem mergeLabels_continuous [TopologicalSpace H] (a : Γ → S → H) :
-    Continuous (fun b : Γ → T → H => mergeLabels a b) := by
-  apply continuous_pi
-  intro g
-  apply continuous_pi
-  intro s
-  cases s with
-  | inl s => exact continuous_const
-  | inr t => exact continuous_apply_apply g t
-
-/-- Copy arbitrary prescribed labels to a finite quotient solely on the finite
-set of vertices currently being tested. This does not assert that the original
-labeling is periodic or descends to a single quotient. -/
- theorem copy_labels_on_finite_set {D : Type*} [Group D] (q : Γ →* D)
-    (P : Finset Γ) (hq : Set.InjOn q (P : Set Γ)) (a : Γ → S → H)
-    (Allowed : H → Prop) (h1 : Allowed 1) (ha : ∀ g s, Allowed (a g s)) :
-    ∃ aF : D → S → H, (∀ d s, Allowed (aF d s)) ∧ ∀ p ∈ P, ∀ s, aF (q p) s = a p s := by
-  classical
-  let aF : D → S → H := fun d s => if hd : ∃ p ∈ P, q p = d then a hd.choose s else 1
-  refine ⟨aF, ?_, ?_⟩
-  · intro d s
-    dsimp [aF]
-    split_ifs
-    · exact ha _ _
-    · exact h1
-  · intro p hp s
-    have he : ∃ p' ∈ P, q p' = q p := ⟨p, hp, rfl⟩
-    simp only [aF, dite_eq_left he]
-    have heq : he.choose = p := hq he.choose_spec.1 hp he.choose_spec.2
-    rw [heq]
-
-/-- Tychonoff compactness for all extra generators, with the original labels
-fixed. The finite-quotient premise is exactly the separate finite solvability
-obligation, not a periodicity assumption on the prescribed assignment. -/
- theorem compact_labeling_from_finite_quotients [Group.ResiduallyFinite Γ]
-    [TopologicalSpace H] [IsTopologicalGroup H] [CompactSpace H]
-    (v : S ⊕ T → Γ) (a : Γ → S → H) (w : I → FreeGroup (S ⊕ T)) (g : I → Γ)
-    (K : I → Set H) (hK : ∀ i, IsClosed (K i))
-    (Allowed : H → Prop) (h1 : Allowed 1) (ha : ∀ g s, Allowed (a g s))
-    (hsolve : ∀ (N : FiniteIndexNormalSubgroup Γ) (J : Finset I)
-      (aF : (Γ ⧸ N.toSubgroup) → S → H), (∀ d s, Allowed (aF d s)) →
-      ∃ bF : (Γ ⧸ N.toSubgroup) → T → H, ∀ i ∈ J, ∀ d,
-        pathProduct (fun s => QuotientGroup.mk' N.toSubgroup (v s))
-          (mergeLabels aF bF) (w i) d ∈ K i) :
-    ∃ b : Γ → T → H, ∀ i, pathProduct v (mergeLabels a b) (w i) (g i) ∈ K i := by
-  classical
-  let C : I → Set (Γ → T → H) := fun i =>
-    {b | pathProduct v (mergeLabels a b) (w i) (g i) ∈ K i}
-  have hc : ∀ i, IsClosed (C i) := fun i =>
-    (hK i).preimage ((pathProduct_continuous v (w i) (g i)).comp (mergeLabels_continuous a))
-  have hf : ∀ J : Finset I, (⋂ i ∈ J, C i).Nonempty := by
-    intro J
-    choose P hP using (fun i : I => pathProduct_finite_locations (H := H) v (w i) (g i))
-    let V := J.biUnion P
-    obtain ⟨N, hN⟩ := finite_set_quotient_injective V
-    let q := QuotientGroup.mk' N.toSubgroup
-    obtain ⟨aF, haF, heF⟩ := copy_labels_on_finite_set q V hN a Allowed h1 ha
-    obtain ⟨bF, hbF⟩ := hsolve N J aF haF
-    refine ⟨fun p t => bF (q p) t, ?_⟩
-    simp only [Set.mem_iInter]
-    intro i hi
-    change pathProduct v (mergeLabels a (fun p t => bF (q p) t)) (w i) (g i) ∈ K i
-    have he : pathProduct v (mergeLabels a (fun p t => bF (q p) t)) (w i) (g i) =
-        pathProduct v (fun p s => mergeLabels aF bF (q p) s) (w i) (g i) := by
-      apply hP i
-      intro p hp s
-      cases s with
-      | inl s => exact (heF p (Finset.mem_biUnion.mpr ⟨i, hi, hp⟩) s).symm
-      | inr t => rfl
-    rw [he, pathProduct_pullback]
-    exact hbF i hi (q (g i))
-  obtain ⟨b, hb⟩ := CompactSpace.iInter_nonempty hc hf
-  exact ⟨b, fun i => Set.mem_iInter.mp hb i⟩
-end CompactLabeling
-end EilenbergGanea
-
-open scoped Quaternion
-namespace EilenbergGanea
-
-theorem su2_norm (a : SU2) : ‖(a : ℍ)‖ = 1 := by
-  have h : star (a : ℍ) * (a : ℍ) = 1 := a.property.1
-  have hn := congrArg norm h
-  simp only [norm_mul, norm_star, norm_one] at hn
-  nlinarith [norm_nonneg (a : ℍ)]
-
-/-- The two elementary estimates needed for the presentation argument, with
-constants independent of the Cayley graph or any finite quotient. -/
-theorem unit_product_linearization (l : List ℍ) (h : ℝ) (hh : 0 ≤ h)
-    (hu : ∀ a ∈ l, ‖a‖ = 1) (hb : ∀ a ∈ l, ‖a - 1‖ ≤ h) :
-    ‖l.prod - 1‖ ≤ l.length * h ∧
-    ‖l.prod - 1 - (l.map (fun a => a - 1)).sum‖ ≤ (l.length : ℝ)^2 * h^2 := by
-  induction l with
-  | nil => simp
-  | cons a l ih =>
-      obtain ⟨hp, he⟩ := ih (fun b hm => hu b (by simp [hm])) (fun b hm => hb b (by simp [hm]))
-      have hna := hu a (by simp)
-      have hba := hb a (by simp)
-      have hid : a * l.prod - 1 = a * (l.prod - 1) + (a - 1) := by noncomm_ring
-      have hie : a * l.prod - 1 - ((a - 1) + (l.map (fun b => b - 1)).sum) =
-          (a - 1) * (l.prod - 1) + (l.prod - 1 - (l.map (fun b => b - 1)).sum) := by noncomm_ring
-      constructor
-      · simp only [List.prod_cons, List.length_cons, Nat.cast_add, Nat.cast_one]
-        rw [hid]
-        calc
-          _ ≤ ‖a * (l.prod - 1)‖ + ‖a - 1‖ := norm_add_le _ _
-          _ = ‖l.prod - 1‖ + ‖a - 1‖ := by rw [norm_mul, hna, one_mul]
-          _ ≤ (l.length + 1) * h := by nlinarith
-      · simp only [List.prod_cons, List.map_cons, List.sum_cons, List.length_cons, Nat.cast_add, Nat.cast_one]
-        rw [hie]
-        have hn : 0 ≤ (l.length : ℝ) := Nat.cast_nonneg _
-        calc
-          _ ≤ ‖(a - 1) * (l.prod - 1)‖ + ‖l.prod - 1 - (l.map (fun b => b - 1)).sum‖ := norm_add_le _ _
-          _ = ‖a - 1‖ * ‖l.prod - 1‖ + ‖l.prod - 1 - (l.map (fun b => b - 1)).sum‖ := by rw [norm_mul]
-          _ ≤ h * (l.length * h) + (l.length : ℝ)^2 * h^2 :=
-            add_le_add (mul_le_mul hba hp (norm_nonneg _) hh) he
-          _ ≤ (l.length + 1)^2 * h^2 := by nlinarith [sq_nonneg h]
-
- theorem quaternion_commutator_bound (a b : ℍ) :
-    ‖a * b - b * a‖ ≤ 2 * ‖a - 1‖ * ‖b - 1‖ := by
-  have he : a * b - b * a = (a - 1) * (b - 1) - (b - 1) * (a - 1) := by noncomm_ring
-  rw [he]
-  calc
-    _ ≤ ‖(a - 1) * (b - 1)‖ + ‖(b - 1) * (a - 1)‖ := norm_sub_le _ _
-    _ = _ := by rw [norm_mul, norm_mul]; ring
-
- theorem su2_conjugation_error (a b : SU2) :
-    ‖((a * b * a⁻¹ : SU2) : ℍ) - (b : ℍ)‖ ≤ 2 * ‖(a : ℍ) - 1‖ * ‖(b : ℍ) - 1‖ := by
-  have he : ((a * b * a⁻¹ : SU2) : ℍ) - (b : ℍ) =
-      ((a : ℍ) * (b : ℍ) - (b : ℍ) * (a : ℍ)) * ((a⁻¹ : SU2) : ℍ) := by
-    change (a : ℍ) * (b : ℍ) * star (a : ℍ) - (b : ℍ) = _
-    change _ = ((a : ℍ) * (b : ℍ) - (b : ℍ) * (a : ℍ)) * star (a : ℍ)
-    rw [sub_mul, mul_assoc (b : ℍ), a.property.2, mul_one]
-  rw [he, norm_mul, su2_norm, mul_one]
-  exact quaternion_commutator_bound _ _
-
- theorem su2_inverse_distance (b : SU2) : ‖((b⁻¹ : SU2) : ℍ) - 1‖ = ‖(b : ℍ) - 1‖ := by
-  change ‖star (b : ℍ) - 1‖ = _
-  calc
-    _ = ‖star ((b : ℍ) - 1)‖ := by rw [star_sub, star_one]
-    _ = _ := norm_star _
-
- theorem su2_inverse_linearization (b : SU2) :
-    ‖((b⁻¹ : SU2) : ℍ) - 1 + ((b : ℍ) - 1)‖ ≤ ‖(b : ℍ) - 1‖^2 := by
-  have he : ((b⁻¹ : SU2) : ℍ) - 1 + ((b : ℍ) - 1) =
-      -((((b⁻¹ : SU2) : ℍ) - 1) * ((b : ℍ) - 1)) := by
-    have hb : ((b⁻¹ : SU2) : ℍ) * (b : ℍ) = 1 := b.property.1
-    noncomm_ring [hb]
-  rw [he, norm_neg, norm_mul, su2_inverse_distance, pow_two]
-
-/-- Replacing each displacement by its linear model introduces only the sum
-of the errors. Combined with exact signed counts, this avoids any choice of
-an ordering of the relator indices (which can be uncountable). -/
-theorem unit_product_with_linear_models {I : Type*} (l : List I)
-    (f : I → SU2) (v : I → ℍ) (h D : ℝ) (hh : 0 ≤ h)
-    (hb : ∀ i ∈ l, ‖(f i : ℍ) - 1‖ ≤ h)
-    (he : ∀ i ∈ l, ‖(f i : ℍ) - 1 - v i‖ ≤ D) :
-    ‖((l.map (fun i => (f i : ℍ))).prod - 1) - (l.map v).sum‖ ≤
-      (l.length : ℝ)^2 * h^2 + l.length * D := by
-  have hp := (unit_product_linearization (l.map (fun i => (f i : ℍ))) h hh
-    (by intro a ha; obtain ⟨i, _, rfl⟩ := List.mem_map.mp ha; exact su2_norm _)
-    (by intro a ha; obtain ⟨i, hi, rfl⟩ := List.mem_map.mp ha; exact hb i hi)).2
-  simp only [List.length_map, List.map_map, Function.comp_def] at hp
-  have hs : ∀ l : List I, (∀ i ∈ l, ‖(f i : ℍ) - 1 - v i‖ ≤ D) →
-      ‖(l.map (fun i => (f i : ℍ) - 1)).sum - (l.map v).sum‖ ≤ l.length * D := by
-    intro l he
-    induction l with
-    | nil => simp
-    | cons i l ih =>
-        have hi := he i (by simp)
-        have hrest := ih (fun j hj => he j (by simp [hj]))
-        simp only [List.map_cons, List.sum_cons, List.length_cons, Nat.cast_add, Nat.cast_one]
-        have heq : (f i : ℍ) - 1 + (l.map (fun j => (f j : ℍ) - 1)).sum -
-            (v i + (l.map v).sum) = ((f i : ℍ) - 1 - v i) +
-              ((l.map (fun j => (f j : ℍ) - 1)).sum - (l.map v).sum) := by abel
-        rw [heq]
-        exact (norm_add_le _ _).trans (by linarith)
-
-  calc
-    _ ≤ ‖((l.map (fun i => (f i : ℍ))).prod - 1) - (l.map (fun i => (f i : ℍ) - 1)).sum‖ +
-      ‖(l.map (fun i => (f i : ℍ) - 1)).sum - (l.map v).sum‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
-    _ ≤ _ := add_le_add hp (hs l he)
-end EilenbergGanea
-
-namespace EilenbergGanea
-open scoped Quaternion
-
-theorem su2_conjugation_distance (a b : SU2) :
-    ‖((a * b * a⁻¹ : SU2) : ℍ) - 1‖ = ‖(b : ℍ) - 1‖ := by
-  have he : ((a * b * a⁻¹ : SU2) : ℍ) - 1 =
-      (a : ℍ) * ((b : ℍ) - 1) * star (a : ℍ) := by
-    change (a : ℍ) * (b : ℍ) * star (a : ℍ) - 1 = _
-    rw [mul_sub, sub_mul, mul_one, a.property.2]
-  rw [he, norm_mul, norm_mul, norm_star, su2_norm, one_mul, mul_one]
-
-def signedUnit (b : SU2) (ε : Bool) : SU2 := if ε then b else b⁻¹
-def signedDisplacement (b : SU2) (ε : Bool) : ℍ := if ε then (b : ℍ) - 1 else -((b : ℍ) - 1)
-
- theorem signedUnit_distance (b : SU2) (ε : Bool) :
-    ‖(signedUnit b ε : ℍ) - 1‖ = ‖(b : ℍ) - 1‖ := by
-  cases ε <;> simp [signedUnit, su2_inverse_distance]
-
- theorem signedUnit_error (b : SU2) (ε : Bool) :
-    ‖(signedUnit b ε : ℍ) - 1 - signedDisplacement b ε‖ ≤ ‖(b : ℍ) - 1‖^2 := by
-  cases ε with
-  | false =>
-      change ‖((b⁻¹ : SU2) : ℍ) - 1 - (-((b : ℍ) - 1))‖ ≤ _
-      simpa only [sub_neg_eq_add] using su2_inverse_linearization b
-  | true => simp [signedUnit, signedDisplacement]
-
- theorem conjugated_signedUnit_error (a b : SU2) (ε : Bool) (A h : ℝ)
-    (hA : 0 ≤ A) (hh : 0 ≤ h) (ha : ‖(a : ℍ) - 1‖ ≤ A) (hb : ‖(b : ℍ) - 1‖ ≤ h) :
-    ‖((a * signedUnit b ε * a⁻¹ : SU2) : ℍ) - 1 - signedDisplacement b ε‖ ≤
-      2 * A * h + h^2 := by
-  have he : ((a * signedUnit b ε * a⁻¹ : SU2) : ℍ) - 1 - signedDisplacement b ε =
-      (((a * signedUnit b ε * a⁻¹ : SU2) : ℍ) - (signedUnit b ε : ℍ)) +
-      ((signedUnit b ε : ℍ) - 1 - signedDisplacement b ε) := by abel
-  rw [he]
-  have hsq : ‖(b : ℍ) - 1‖^2 ≤ h^2 := by nlinarith [norm_nonneg ((b : ℍ) - 1)]
-  calc
-    _ ≤ ‖((a * signedUnit b ε * a⁻¹ : SU2) : ℍ) - (signedUnit b ε : ℍ)‖ +
-      ‖(signedUnit b ε : ℍ) - 1 - signedDisplacement b ε‖ := norm_add_le _ _
-    _ ≤ 2 * ‖(a : ℍ) - 1‖ * ‖(b : ℍ) - 1‖ + ‖(b : ℍ) - 1‖^2 := by
-      simpa only [signedUnit_distance] using add_le_add (su2_conjugation_error a (signedUnit b ε)) (signedUnit_error b ε)
-    _ ≤ _ := add_le_add (mul_le_mul (mul_le_mul_of_nonneg_left ha (by positivity)) hb
-      (norm_nonneg _) (mul_nonneg (by positivity) hA)) hsq
-
-abbrev RelatorFactor (I : Type*) := SU2 × I × Bool
-
-def relatorFactorValue {I : Type*} (x : I → SU2) (r : RelatorFactor I) : SU2 :=
-  r.1 * signedUnit (x r.2.1) r.2.2 * r.1⁻¹
-
-def relatorSignedCounts {I : Type*} (l : List (RelatorFactor I)) : I →₀ ℤ :=
-  (l.map (fun r => Finsupp.single r.2.1 (if r.2.2 then 1 else -1))).sum
-
- theorem relator_linear_models {I : Type*} (l : List (RelatorFactor I)) (x : I → SU2) :
-    (l.map (fun r => signedDisplacement (x r.2.1) r.2.2)).sum =
-      Finsupp.linearCombination ℤ (fun i => (x i : ℍ) - 1) (relatorSignedCounts l) := by
-  induction l with
-  | nil => simp [relatorSignedCounts]
-  | cons r l ih =>
-      simp only [List.map_cons, List.sum_cons, relatorSignedCounts, map_add, Finsupp.linearCombination_single]
-      rw [← show Finsupp.linearCombination ℤ (fun i => (x i : ℍ) - 1)
-          ((l.map (fun r => Finsupp.single r.2.1 (if r.2.2 then 1 else -1))).sum) =
-          (l.map (fun r => signedDisplacement (x r.2.1) r.2.2)).sum from ih.symm]
-      congr 2
-      cases r.2.2 <;> simp [signedDisplacement]
-
-/-- Exact translated counts cancel the full linear term; the remaining bound
-is quadratic. The index set is arbitrary, not necessarily countable. -/
-theorem relator_product_quadratic {I : Type*} (l : List (RelatorFactor I)) (x : I → SU2)
-    (i₀ : I) (hc : relatorSignedCounts l = Finsupp.single i₀ 1)
-    (h ε K : ℝ) (hh : 0 ≤ h) (hε : 0 ≤ ε) (hK : 0 ≤ K)
-    (hb : ∀ r ∈ l, ‖(x r.2.1 : ℍ) - 1‖ ≤ h)
-    (ha : ∀ r ∈ l, ‖(r.1 : ℍ) - 1‖ ≤ K * (ε + h)) :
-    ‖(l.map (fun r => (relatorFactorValue x r : ℍ))).prod - (x i₀ : ℍ)‖ ≤
-      ((l.length : ℝ)^2 + l.length + 2 * l.length * K) * (ε + h) * h := by
-  have hp := unit_product_with_linear_models l (relatorFactorValue x)
-    (fun r => signedDisplacement (x r.2.1) r.2.2) h
-    (2 * (K * (ε + h)) * h + h^2) hh
-    (fun r hr => by rw [relatorFactorValue, su2_conjugation_distance, signedUnit_distance]; exact hb r hr)
-    (fun r hr => conjugated_signedUnit_error r.1 (x r.2.1) r.2.2 _ h
-      (mul_nonneg hK (add_nonneg hε hh)) hh (ha r hr) (hb r hr))
-  rw [relator_linear_models, hc, Finsupp.linearCombination_single, one_smul] at hp
-  have he : (l.map (fun r => (relatorFactorValue x r : ℍ))).prod - 1 - ((x i₀ : ℍ) - 1) =
-      (l.map (fun r => (relatorFactorValue x r : ℍ))).prod - (x i₀ : ℍ) := by abel
-  rw [he] at hp
-  apply hp.trans
-  have hlen : 0 ≤ (l.length : ℝ) := Nat.cast_nonneg _
-  have hnh : 0 ≤ ((l.length : ℝ)^2 + l.length) * ε * h :=
-    mul_nonneg (mul_nonneg (add_nonneg (sq_nonneg _) hlen) hε) hh
-  nlinarith
-end EilenbergGanea
-
-namespace EilenbergGanea
-
-/-- A uniform size for an arbitrary family, normalized to zero for an empty
-index set. This is a real supremum, not an attained maximum. -/
-def familySize {I : Type*} (f : I → ℝ) : ℝ := sSup (Set.range f ∪ {0})
-
- theorem familySize_nonneg {I : Type*} (f : I → ℝ) (hf : BddAbove (Set.range f)) :
-    0 ≤ familySize f := by
-  apply le_csSup (hf.union bddAbove_singleton)
-  exact Or.inr (Set.mem_singleton _)
-
- theorem le_familySize {I : Type*} (f : I → ℝ) (hf : BddAbove (Set.range f)) (i : I) :
-    f i ≤ familySize f := le_csSup (hf.union bddAbove_singleton) (Or.inl (Set.mem_range_self i))
-
- theorem familySize_le {I : Type*} (f : I → ℝ) {c : ℝ} (hc : 0 ≤ c) (hf : ∀ i, f i ≤ c) :
-    familySize f ≤ c := by
-  apply csSup_le (Set.union_nonempty.mpr (Or.inr (Set.singleton_nonempty 0)))
-  rintro x (⟨i, rfl⟩ | rfl)
-  · exact hf i
-  · exact hc
-
-/-- The source no-threshold-crossing argument does not assume attainment of
-its supremum. -/
- theorem familySize_ne_threshold {I : Type*} (f : I → ℝ)
-    (hf : BddAbove (Set.range f)) (C δ η ε : ℝ)
-    (hδ : 0 < δ) (hC : 0 ≤ C) (_hη : 0 ≤ η) (hε : 0 ≤ ε) (hεη : ε ≤ η)
-    (hc₁ : C * (δ + η) < 1/2) (hc₂ : C * η < δ/2)
-    (hbound : ∀ i, f i ≤ C * ε + C * (familySize f + ε) * familySize f) :
-    familySize f ≠ δ := by
-  intro he
-  have hs : familySize f ≤ C * ε + C * (familySize f + ε) * familySize f :=
-    familySize_le f (by positivity [familySize_nonneg f hf]) hbound
-  rw [he] at hs
-  have hce : C * ε ≤ C * η := mul_le_mul_of_nonneg_left hεη hC
-  have hfactor : C * (δ + ε) ≤ C * (δ + η) := mul_le_mul_of_nonneg_left (by linarith) hC
-  nlinarith [mul_lt_mul_of_pos_right hc₁ hδ, mul_le_mul_of_nonneg_right hfactor hδ.le]
-
-/-- Uniform contraction of all residual errors. This handles even an
-uncountable family and does not replace the supremum by a maximum. -/
- theorem familySize_contraction {I : Type*} (f : I → ℝ)
-    (hf : ∀ i, 0 ≤ f i) (δ C η : ℝ) (hδ : 0 ≤ δ) (hC : 0 ≤ C) (hη : 0 ≤ η)
-    (hb : ∀ i, f i ≤ δ) (hc : C * (δ + η) < 1)
-    (he : ∀ i, f i ≤ C * (familySize f + η) * familySize f) : ∀ i, f i = 0 := by
-  have hbounded : BddAbove (Set.range f) := ⟨δ, by rintro _ ⟨i, rfl⟩; exact hb i⟩
-  have h0 := familySize_nonneg f hbounded
-  have hmax := familySize_le f hδ hb
-  have hs := familySize_le f (c := C * (familySize f + η) * familySize f) (by positivity) he
-  have hmul : C * (familySize f + η) ≤ C * (δ + η) :=
-    mul_le_mul_of_nonneg_left (by linarith) hC
-  have heq : familySize f = 0 := by
-    by_contra hn
-    have hp : 0 < familySize f := lt_of_le_of_ne h0 (Ne.symm hn)
-    have ht := mul_lt_mul_of_pos_right (lt_of_le_of_lt hmul hc) hp
-    nlinarith
-  intro i
-  exact le_antisymm (by simpa only [heq] using le_familySize f hbounded i) (hf i)
-
- theorem small_thresholds_exist (C : ℝ) (hC : 1 ≤ C) :
-    ∃ δ η : ℝ, 0 < δ ∧ δ < 1 ∧ 0 < η ∧ η < 1 ∧
-      C * (δ + η) < 1/2 ∧ C * η < δ/2 := by
-  have hc : 0 < C := by linarith
-  let δ := 1 / (8*C)
-  let η := δ / (8*C)
-  have hd : 0 < δ := by dsimp [δ]; positivity
-  have hd1 : δ < 1 := by dsimp [δ]; apply (div_lt_iff₀ (by positivity)).mpr; linarith
-  have he : 0 < η := by dsimp [η]; positivity
-  have heδ : η < δ := by dsimp [η]; apply (div_lt_iff₀ (by positivity)).mpr; nlinarith
-  have hcd : C * δ = 1/8 := by dsimp [δ]; field_simp
-  have hce : C * η = δ/8 := by dsimp [η]; field_simp
-  refine ⟨δ, η, hd, hd1, he, heδ.trans hd1, ?_, ?_⟩
-  · rw [mul_add, hcd, hce]; linarith
-  · rw [hce]; linarith
-end EilenbergGanea
-
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.WordEquations
+import OAI.GroupTheory.RightAngledArtin.ResidualFiniteness
+import Mathlib.Topology.Compactness.Compact
+import Mathlib.LinearAlgebra.Finsupp.LinearCombination
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set Filter Topology MeasureTheory
+open scoped Quaternion ContDiff
+
+namespace EilenbergGanea
+open scoped Quaternion ContDiff
+/-- Squared chord length on the unit-quaternion sphere. -/
+theorem su2_distance_square (a : SU2) :
+    ‖(a : ℍ) - 1‖ ^ 2 = 2 - 2 * (a : ℍ).re := by
+  have hu := (quaternion_mem_unitary_iff _).mp a.property
+  have hs : Quaternion.normSq ((a : ℍ) - 1) = Quaternion.normSq (a : ℍ) + 1 - 2 * (a : ℍ).re := by
+    simp only [Quaternion.normSq_def', Quaternion.re_sub, Quaternion.imI_sub,
+      Quaternion.imJ_sub, Quaternion.imK_sub, Quaternion.re_one,
+      Quaternion.imI_one, Quaternion.imJ_one, Quaternion.imK_one]
+    ring
+  simpa only [Quaternion.normSq_eq_norm_mul_self, hu, one_mul, sq, one_add_one_eq_two] using hs
+
+def arcBlend (a : SU2) (t : ℝ) : ℍ :=
+  (1 - Real.smoothTransition t) • (1 : ℍ) + Real.smoothTransition t • (a : ℍ)
+
+theorem arcBlend_norm_le (a : SU2) (t : ℝ) : ‖arcBlend a t‖ ≤ 1 := by
+  have h0 := Real.smoothTransition.nonneg t
+  have h1 := Real.smoothTransition.le_one t
+  have hn := (quaternion_mem_unitary_iff _).mp a.property
+  calc
+    _ ≤ ‖(1 - Real.smoothTransition t) • (1 : ℍ)‖ +
+        ‖Real.smoothTransition t • (a : ℍ)‖ := norm_add_le _ _
+    _ = 1 := by rw [norm_smul, norm_smul, norm_one, hn, Real.norm_eq_abs,
+        Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr h1), abs_of_nonneg h0]; ring
+
+theorem small_unit_re (a : SU2) (ha : ‖(a : ℍ) - 1‖ < 1) :
+    0 < (a : ℍ).re ∧ (a : ℍ).re ≤ 1 := by
+  have h := su2_distance_square a
+  have hn := norm_nonneg ((a : ℍ) - 1)
+  constructor <;> nlinarith [sq_nonneg ‖(a : ℍ) - 1‖]
+
+theorem arcBlend_re_le (a : SU2) (ha : ‖(a : ℍ) - 1‖ < 1) (t : ℝ) :
+    (a : ℍ).re ≤ (arcBlend a t).re := by
+  have h := (small_unit_re a ha).2
+  have h0 := Real.smoothTransition.nonneg t
+  have h1 := Real.smoothTransition.le_one t
+  simp only [arcBlend, Quaternion.re_add, Quaternion.re_smul, Quaternion.re_one, smul_eq_mul]
+  nlinarith
+
+theorem arcBlend_ne_zero (a : SU2) (ha : ‖(a : ℍ) - 1‖ < 1) (t : ℝ) :
+    arcBlend a t ≠ 0 := by
+  have h := (small_unit_re a ha).1.trans_le (arcBlend_re_le a ha t)
+  intro he
+  simp only [he, Quaternion.re_zero, lt_self_iff_false] at h
+
+/-- A globally smooth path that remains in the closed chord ball of its
+endpoint, avoiding any branch choice of a quaternion logarithm. -/
+def smallQuaternionArc (a : SU2) (t : ℝ) : SU2 := normalizeQuaternion (arcBlend a t)
+
+@[simp] theorem smallQuaternionArc_zero (a : SU2) : smallQuaternionArc a 0 = 1 := by
+  simpa [smallQuaternionArc, arcBlend] using (normalizeQuaternion_unitary (1 : SU2))
+
+@[simp] theorem smallQuaternionArc_one (a : SU2) : smallQuaternionArc a 1 = a := by
+  simp [smallQuaternionArc, arcBlend]
+
+theorem smallQuaternionArc_smooth (a : SU2) (ha : ‖(a : ℍ) - 1‖ < 1) :
+    ContDiff ℝ ∞ (fun t => (smallQuaternionArc a t : ℍ)) := by
+  apply contDiff_iff_contDiffAt.mpr
+  intro t
+  exact (normalizeQuaternion_coe_contDiffAt (arcBlend_ne_zero a ha t)).comp t
+    (((contDiff_const.sub Real.smoothTransition.contDiff).smul contDiff_const).add
+      (Real.smoothTransition.contDiff.smul contDiff_const)).contDiffAt
+
+theorem smallQuaternionArc_distance (a : SU2) (ha : ‖(a : ℍ) - 1‖ < 1) (t : ℝ) :
+    ‖(smallQuaternionArc a t : ℍ) - 1‖ ≤ ‖(a : ℍ) - 1‖ := by
+  have hp : 0 < ‖arcBlend a t‖ := norm_pos_iff.mpr (arcBlend_ne_zero a ha t)
+  have hre : (a : ℍ).re ≤ (smallQuaternionArc a t : ℍ).re := by
+    rw [smallQuaternionArc, normalizeQuaternion_coe_of_ne_zero (arcBlend_ne_zero a ha t)]
+    simp only [Quaternion.re_smul, smul_eq_mul]
+    rw [mul_comm, ← div_eq_mul_inv, le_div_iff₀ hp]
+    exact (mul_le_of_le_one_right (small_unit_re a ha).1.le (arcBlend_norm_le a t)).trans
+      (arcBlend_re_le a ha t)
+  have h0 := su2_distance_square a
+  have h1 := su2_distance_square (smallQuaternionArc a t)
+  nlinarith [norm_nonneg ((a : ℍ)-1), norm_nonneg ((smallQuaternionArc a t : ℍ)-1)]
+
+end EilenbergGanea
+
+namespace EilenbergGanea
+open scoped Quaternion ContDiff
+open Set
+section SelectedEquations
+variable {E : Type*} {m d : ℕ}
+
+/-- The selected unknowns remain separate from coefficients, even when two
+letters have the same image in the group quotient. -/
+def selectedAlphabet (σ : Fin m ↪ E) (e : E) : E ⊕ Fin m := by
+  classical
+  exact if h : ∃ i, σ i = e then .inr h.choose else .inl e
+
+@[simp] theorem selectedAlphabet_selected (σ : Fin m ↪ E) (i : Fin m) :
+    selectedAlphabet σ (σ i) = .inr i := by
+  classical
+  have h : ∃ j, σ j = σ i := ⟨i,rfl⟩
+  simp only [selectedAlphabet, dite_eq_left h]
+  exact congrArg Sum.inr (σ.injective h.choose_spec)
+
+ theorem selectedAlphabet_other (σ : Fin m ↪ E) (e : E) (he : ∀ i, σ i ≠ e) :
+    selectedAlphabet σ e = .inl e := by
+  classical
+  simp [selectedAlphabet, he]
+
+def selectedWord (σ : Fin m ↪ E) : FreeGroup E →* FreeGroup (E ⊕ Fin m) :=
+  FreeGroup.map (selectedAlphabet σ)
+
+def selectedLabel (σ : Fin m ↪ E) (a : E → SU2) (x : Fin m → SU2) (e : E) : SU2 :=
+  Sum.elim a x (selectedAlphabet σ e)
+
+@[simp] theorem selectedLabel_selected (σ : Fin m ↪ E) (a : E → SU2)
+    (x : Fin m → SU2) (i : Fin m) : selectedLabel σ a x (σ i) = x i := by
+  simp [selectedLabel]
+
+ theorem selectedLabel_other (σ : Fin m ↪ E) (a : E → SU2) (x : Fin m → SU2)
+    (e : E) (he : ∀ i, σ i ≠ e) : selectedLabel σ a x e = a e := by
+  rw [selectedLabel, selectedAlphabet_other σ e he]
+  rfl
+
+ theorem selectedWord_eval (σ : Fin m ↪ E) (a : E → SU2) (x : Fin m → SU2)
+    (w : FreeGroup E) :
+    FreeGroup.lift (Sum.elim a x) (selectedWord σ w) =
+      FreeGroup.lift (selectedLabel σ a x) w := by
+  have he : (FreeGroup.lift (Sum.elim a x)).comp (selectedWord σ) =
+      FreeGroup.lift (selectedLabel σ a x) := by
+    apply FreeGroup.ext_hom
+    intro e
+    simp [selectedWord, selectedLabel]
+  exact congrArg (fun f : FreeGroup E →* SU2 => f w) he
+
+ theorem selectedWord_exponents [DecidableEq E] (σ : Fin m ↪ E) (w : FreeGroup E) (j : Fin m) :
+    wordExponents (eraseCoefficients (selectedWord σ w)) j = wordExponents w (σ j) := by
+  classical
+  have he : (FreeGroup.lift (fun i : Fin m => Multiplicative.ofAdd (if j = i then (1 : ℤ) else 0))).comp
+      (eraseCoefficients.comp (selectedWord σ)) =
+      FreeGroup.lift (fun e : E => Multiplicative.ofAdd (if σ j = e then (1 : ℤ) else 0)) := by
+    apply FreeGroup.ext_hom
+    intro e
+    by_cases h : ∃ i, σ i = e
+    · obtain ⟨i,rfl⟩ := h
+      simp [selectedWord, eraseCoefficients, σ.injective.eq_iff]
+    · have h' : ∀ i, σ i ≠ e := by simpa only [not_exists] using h
+      simp [selectedWord, eraseCoefficients, selectedAlphabet_other σ e h', h' j]
+  exact congrArg (fun f : FreeGroup E →* Multiplicative ℤ => (f w).toAdd) he
+
+/-- Solve a finite, full-rank system while keeping every unselected label fixed.
+The distinguished sublevel condition is the actual isolated root condition,
+not a nonsingularity assumption on the roots. -/
+theorem selected_word_sublevel_solvable [DecidableEq E]
+    (σ : Fin m ↪ E) (w : Fin m → FreeGroup E) (ρ : Fin d → FreeGroup E)
+    (a : ℝ → E → SU2) (ha : ∀ e, ContDiff ℝ ∞ (fun t => (a t e : ℍ)))
+    (ha0 : ∀ e, a 0 e = 1)
+    (hB : Matrix.det (fun i j => wordExponents (w i) (σ j)) ≠ 0)
+    (δ : ℝ) (hδ : 0 < δ)
+    (hiso : ∀ t ∈ Icc (0 : ℝ) 1, ∀ x : Fin m → SU2,
+      (∀ i, FreeGroup.lift (selectedLabel σ (a t) x) (w i) = 1) →
+      ‖(fun i => (FreeGroup.lift (selectedLabel σ (a t) x) (ρ i) : ℍ)) - 1‖ ≠ δ) :
+    ∃ b : E → SU2,
+      (∀ e, (∀ i, σ i ≠ e) → b e = a 1 e) ∧
+      (∀ i, FreeGroup.lift b (w i) = 1) ∧
+      ‖(fun i => (FreeGroup.lift b (ρ i) : ℍ)) - 1‖ < δ := by
+  have hb : (exponentMatrix (fun i => eraseCoefficients (selectedWord σ (w i)))).det ≠ 0 := by
+    change (Matrix.det (fun i j => wordExponents (eraseCoefficients (selectedWord σ (w i))) j)) ≠ 0
+    simpa only [selectedWord_exponents] using hB
+  obtain ⟨x,hx,hρ⟩ := coefficient_word_sublevel_solvable
+    (fun i => selectedWord σ (w i)) (fun i => selectedWord σ (ρ i)) a ha ha0 hb δ hδ
+    (fun t ht x hx => by
+      simp only [coefficientWordMap, selectedWord_eval]
+      apply hiso t ht x
+      intro i
+      have hi := congrFun hx i
+      simpa only [coefficientWordMap, selectedWord_eval, Pi.one_apply] using hi)
+  refine ⟨selectedLabel σ (a 1) x, fun e he => selectedLabel_other σ _ x e he, ?_, ?_⟩
+  · intro i
+    have hi := congrFun hx i
+    simpa only [coefficientWordMap, selectedWord_eval, Pi.one_apply] using hi
+  · simpa only [coefficientWordMap, selectedWord_eval] using hρ
+
+end SelectedEquations
+end EilenbergGanea
+
+namespace EilenbergGanea
+
+variable {Γ S H : Type*} [Group Γ] [Group H]
+
+def labelLift (v : S → Γ) (a : Γ → S → H) :
+    FreeGroup S →* ((Γ → H) ⋊[functionShift Γ H] Γ) :=
+  FreeGroup.lift (fun s => ⟨fun g => a g s, v s⟩)
+
+def wordValue (v : S → Γ) : FreeGroup S →* Γ := FreeGroup.lift v
+
+def pathProduct (v : S → Γ) (a : Γ → S → H) (w : FreeGroup S) (g : Γ) : H :=
+  (labelLift v a w).left g
+
+ theorem labelLift_right (v : S → Γ) (a : Γ → S → H) (w : FreeGroup S) :
+    (labelLift v a w).right = wordValue v w := by
+  have he : (SemidirectProduct.rightHom : ((Γ → H) ⋊[functionShift Γ H] Γ) →* Γ).comp
+      (labelLift v a) = wordValue v := by
+    apply FreeGroup.ext_hom
+    intro s
+    simp [labelLift, wordValue]
+  exact congrArg (fun f : FreeGroup S →* Γ => f w) he
+
+@[simp] theorem pathProduct_one (v : S → Γ) (a : Γ → S → H) (g : Γ) :
+    pathProduct v a 1 g = 1 := by simp [pathProduct]
+@[simp] theorem pathProduct_of (v : S → Γ) (a : Γ → S → H) (s : S) (g : Γ) :
+    pathProduct v a (FreeGroup.of s) g = a g s := by simp [pathProduct, labelLift]
+ theorem pathProduct_mul (v : S → Γ) (a : Γ → S → H) (w z : FreeGroup S) (g : Γ) :
+    pathProduct v a (w*z) g = pathProduct v a w g * pathProduct v a z (g * wordValue v w) := by
+  simp only [pathProduct, map_mul, SemidirectProduct.mul_left, Pi.mul_apply, labelLift_right]
+  rfl
+ theorem pathProduct_inv (v : S → Γ) (a : Γ → S → H) (w : FreeGroup S) (g : Γ) :
+    pathProduct v a w⁻¹ g = (pathProduct v a w (g * (wordValue v w)⁻¹))⁻¹ := by
+  simp only [pathProduct, map_inv, SemidirectProduct.inv_left, labelLift_right]
+  rfl
+
+ theorem pathProduct_conjugate_loop (v : S → Γ) (a : Γ → S → H)
+    (u r : FreeGroup S) (hr : wordValue v r = 1) (g : Γ) :
+    pathProduct v a (u*r*u⁻¹) g = pathProduct v a u g *
+      pathProduct v a r (g * wordValue v u) * (pathProduct v a u g)⁻¹ := by
+  rw [pathProduct_mul, pathProduct_mul, pathProduct_inv, map_mul, hr, mul_one]
+  simp only [mul_inv_cancel_right]
+
+ theorem pathProduct_continuous [TopologicalSpace H] [IsTopologicalGroup H]
+    (v : S → Γ) (w : FreeGroup S) (g : Γ) :
+    Continuous (fun a : Γ → S → H => pathProduct v a w g) := by
+  induction w using FreeGroup.induction_on generalizing g with
+  | one => simpa using (continuous_const : Continuous (fun _ : Γ → S → H => (1 : H)))
+  | of s =>
+      simp only [pathProduct_of]
+      exact continuous_apply_apply g s
+  | inv_of s =>
+      simp only [pathProduct_inv, pathProduct_of]
+      exact (continuous_apply_apply (g * (wordValue v (FreeGroup.of s))⁻¹) s).inv
+  | mul w z hw hz =>
+      simp only [pathProduct_mul]
+      exact (hw g).mul (hz (g * wordValue v w))
+
+/-- The set of vertices inspected by a word is finite, independently of the
+label values. No countability hypothesis on the alphabet is needed. -/
+theorem pathProduct_finite_locations (v : S → Γ) (w : FreeGroup S) (g : Γ) :
+    ∃ P : Finset Γ, ∀ a b : Γ → S → H,
+      (∀ p ∈ P, ∀ s, a p s = b p s) → pathProduct v a w g = pathProduct v b w g := by
+  classical
+  induction w using FreeGroup.induction_on generalizing g with
+  | one => exact ⟨∅, fun _ _ _ => by simp⟩
+  | of s => exact ⟨{g}, fun a b hab => by simpa using hab g (by simp) s⟩
+  | inv_of s =>
+      exact ⟨{g * (wordValue v (FreeGroup.of s))⁻¹}, fun a b hab => by
+        rw [pathProduct_inv, pathProduct_inv, pathProduct_of, pathProduct_of, hab _ (by simp)]⟩
+  | mul w z hw hz =>
+      obtain ⟨P, hP⟩ := hw g
+      obtain ⟨Q, hQ⟩ := hz (g * wordValue v w)
+      refine ⟨P �� Q, fun a b hab => ?_⟩
+      rw [pathProduct_mul, pathProduct_mul, hP a b (fun p hp => hab p (Finset.mem_union_left Q hp)),
+        hQ a b (fun p hp => hab p (Finset.mem_union_right P hp))]
+
+/-- Evaluation commutes with pullback of a labeling along a group quotient,
+including when different alphabet letters have the same image. -/
+theorem pathProduct_pullback {D : Type*} [Group D] (q : Γ →* D)
+    (v : S → Γ) (a : D → S → H) (w : FreeGroup S) (g : Γ) :
+    pathProduct v (fun p s => a (q p) s) w g = pathProduct (fun s => q (v s)) a w (q g) := by
+  have hv : ∀ w : FreeGroup S, wordValue (fun s => q (v s)) w = q (wordValue v w) := by
+    have he : wordValue (fun s => q (v s)) = q.comp (wordValue v) := by
+      apply FreeGroup.ext_hom
+      intro s
+      simp [wordValue]
+    intro w
+    exact congrArg (fun f : FreeGroup S →* D => f w) he
+  induction w using FreeGroup.induction_on generalizing g with
+  | one => simp
+  | of s => simp
+  | inv_of s => simp only [pathProduct_inv, pathProduct_of, hv, map_inv, map_mul]
+  | mul w z hw hz => simp only [pathProduct_mul, hv, map_mul, hw, hz]
+
+
+/-- Once every final relator has trivial transport at every vertex, normal
+closure gives trivial transport on every null word, not just on a finite
+collection of test loops. -/
+ theorem pathProduct_trivial_of_relators (v : S → Γ) (a : Γ → S → H)
+    (R : Set (FreeGroup S)) (hR : R ⊆ (wordValue v).ker)
+    (hp : (wordValue v).ker = Subgroup.normalClosure R)
+    (ha : ∀ r ∈ R, ∀ g, pathProduct v a r g = 1)
+    (w : FreeGroup S) (hw : wordValue v w = 1) (g : Γ) : pathProduct v a w g = 1 := by
+  have hrel : R ⊆ (labelLift v a).ker := by
+    intro r hr
+    change labelLift v a r = 1
+    apply SemidirectProduct.ext
+    · funext g
+      exact ha r hr g
+    · rw [labelLift_right]
+      exact hR hr
+  have hle : Subgroup.normalClosure R ≤ (labelLift v a).ker :=
+    Subgroup.normalClosure_le_normal hrel
+  have he : labelLift v a w = 1 := hle (by rw [← hp]; exact hw)
+  change (labelLift v a w).left g = 1
+  rw [he]
+  rfl
+
+ def pathChain (v : S → Γ) (w : FreeGroup S) (g : Γ) : (Γ × S) →₀ ℤ :=
+  (pathProduct v (fun p s => Multiplicative.ofAdd (Finsupp.single (p,s) 1)) w g).toAdd
+
+@[simp] theorem pathChain_one (v : S → Γ) (g : Γ) : pathChain v 1 g = 0 := by
+  simp [pathChain]
+@[simp] theorem pathChain_of (v : S → Γ) (s : S) (g : Γ) :
+    pathChain v (FreeGroup.of s) g = Finsupp.single (g,s) 1 := by simp [pathChain]
+ theorem pathChain_mul (v : S → Γ) (w z : FreeGroup S) (g : Γ) :
+    pathChain v (w*z) g = pathChain v w g + pathChain v z (g * wordValue v w) := by
+  simp only [pathChain, pathProduct_mul, toAdd_mul]
+ theorem pathChain_inv (v : S → Γ) (w : FreeGroup S) (g : Γ) :
+    pathChain v w⁻¹ g = -pathChain v w (g * (wordValue v w)⁻¹) := by
+  simp only [pathChain, pathProduct_inv, toAdd_inv]
+ theorem pathChain_conjugate (v : S → Γ) (u r : FreeGroup S)
+    (hr : wordValue v r = 1) (g : Γ) :
+    pathChain v (u*r*u⁻¹) g = pathChain v r (g * wordValue v u) := by
+  simp only [pathChain, pathProduct_conjugate_loop v _ u r hr g,
+    toAdd_mul, toAdd_inv]
+  abel
+
+variable {R : Type*}
+abbrev NormalFactor (S R : Type*) := FreeGroup S × R × Bool
+
+ def normalFactorWord (r : R → FreeGroup S) (f : NormalFactor S R) : FreeGroup S :=
+  f.1 * (if f.2.2 then r f.2.1 else (r f.2.1)⁻¹) * f.1⁻¹
+
+ def normalFactorCount (v : S → Γ) (f : NormalFactor S R) (g : Γ) : (Γ × R) →₀ ℤ :=
+  Finsupp.single (g * wordValue v f.1, f.2.1) (if f.2.2 then 1 else -1)
+
+ def relatorChainMap (v : S → Γ) (r : R → FreeGroup S) :
+    ((Γ × R) →₀ ℤ) →ₗ[ℤ] ((Γ × S) →₀ ℤ) :=
+  Finsupp.linearCombination ℤ (fun i => pathChain v (r i.2) i.1)
+
+ theorem normalFactor_value (v : S → Γ) (r : R → FreeGroup S)
+    (hr : ∀ i, wordValue v (r i) = 1) (f : NormalFactor S R) :
+    wordValue v (normalFactorWord r f) = 1 := by
+  dsimp only [normalFactorWord]
+  split_ifs <;> simp [hr]
+
+ theorem pathChain_normalFactor (v : S → Γ) (r : R → FreeGroup S)
+    (hr : ∀ i, wordValue v (r i) = 1) (f : NormalFactor S R) (g : Γ) :
+    pathChain v (normalFactorWord r f) g = relatorChainMap v r (normalFactorCount v f g) := by
+  simp only [normalFactorWord, normalFactorCount, relatorChainMap, Finsupp.linearCombination_single]
+  split_ifs with h
+  · rw [pathChain_conjugate v _ _ (hr _)]; simp only [one_smul]
+  · rw [pathChain_conjugate v _ _ (by simp [hr]), pathChain_inv, hr]
+    simp only [inv_one, mul_one, neg_one_smul]
+
+ theorem pathChain_normalProduct (v : S → Γ) (r : R → FreeGroup S)
+    (hr : ∀ i, wordValue v (r i) = 1) (l : List (NormalFactor S R)) (g : Γ) :
+    pathChain v (l.map (normalFactorWord r)).prod g =
+      relatorChainMap v r (l.map (fun f => normalFactorCount v f g)).sum := by
+  induction l with
+  | nil => simp
+  | cons f l ih =>
+      simp only [List.map_cons, List.prod_cons, List.sum_cons, map_add, pathChain_mul,
+        normalFactor_value v r hr f, mul_one, pathChain_normalFactor v r hr f g, ih]
+
+/-- The coefficient identity is an equality in the full free abelian group of
+translated indexed relators. Injectivity is sufficient; no finiteness of the
+relator or generator types is assumed. -/
+ theorem exact_translated_relator_counts (v : S → Γ) (r : R → FreeGroup S)
+    (hr : ∀ i, wordValue v (r i) = 1) (hinj : Function.Injective (relatorChainMap v r))
+    (z : FreeGroup S) (ρ : R) (hchain : pathChain v z 1 = pathChain v (r ρ) 1)
+    (l : List (NormalFactor S R)) (he : z = (l.map (normalFactorWord r)).prod) :
+    (l.map (fun f => normalFactorCount v f 1)).sum = Finsupp.single (1,ρ) 1 := by
+  apply hinj
+  rw [← pathChain_normalProduct v r hr, ← he, hchain]
+  simp [relatorChainMap]
+
+end EilenbergGanea
+
+namespace EilenbergGanea
+variable {Γ S H : Type*} [Group Γ] [Group H]
+
+
+section CompactLabeling
+variable {T I : Type*}
+
+ def mergeLabels (a : Γ → S → H) (b : Γ → T → H) : Γ → S ⊕ T → H :=
+  fun g => Sum.elim (a g) (b g)
+
+ omit [Group Γ] [Group H] in
+ theorem mergeLabels_continuous [TopologicalSpace H] (a : Γ → S → H) :
+    Continuous (fun b : Γ → T → H => mergeLabels a b) := by
+  apply continuous_pi
+  intro g
+  apply continuous_pi
+  intro s
+  cases s with
+  | inl s => exact continuous_const
+  | inr t => exact continuous_apply_apply g t
+
+/-- Copy arbitrary prescribed labels to a finite quotient solely on the finite
+set of vertices currently being tested. This does not assert that the original
+labeling is periodic or descends to a single quotient. -/
+ theorem copy_labels_on_finite_set {D : Type*} [Group D] (q : Γ →* D)
+    (P : Finset Γ) (hq : Set.InjOn q (P : Set Γ)) (a : Γ → S → H)
+    (Allowed : H → Prop) (h1 : Allowed 1) (ha : ∀ g s, Allowed (a g s)) :
+    ∃ aF : D → S → H, (∀ d s, Allowed (aF d s)) ∧ ∀ p ∈ P, ∀ s, aF (q p) s = a p s := by
+  classical
+  let aF : D → S → H := fun d s => if hd : ∃ p ∈ P, q p = d then a hd.choose s else 1
+  refine ⟨aF, ?_, ?_⟩
+  · intro d s
+    dsimp [aF]
+    split_ifs
+    · exact ha _ _
+    · exact h1
+  · intro p hp s
+    have he : ∃ p' ∈ P, q p' = q p := ⟨p, hp, rfl⟩
+    simp only [aF, dite_eq_left he]
+    have heq : he.choose = p := hq he.choose_spec.1 hp he.choose_spec.2
+    rw [heq]
+
+/-- Tychonoff compactness for all extra generators, with the original labels
+fixed. The finite-quotient premise is exactly the separate finite solvability
+obligation, not a periodicity assumption on the prescribed assignment. -/
+ theorem compact_labeling_from_finite_quotients [Group.ResiduallyFinite Γ]
+    [TopologicalSpace H] [IsTopologicalGroup H] [CompactSpace H]
+    (v : S ⊕ T → Γ) (a : Γ → S → H) (w : I → FreeGroup (S ⊕ T)) (g : I → Γ)
+    (K : I → Set H) (hK : ∀ i, IsClosed (K i))
+    (Allowed : H → Prop) (h1 : Allowed 1) (ha : ∀ g s, Allowed (a g s))
+    (hsolve : ∀ (N : FiniteIndexNormalSubgroup Γ) (J : Finset I)
+      (aF : (Γ ⧸ N.toSubgroup) → S → H), (∀ d s, Allowed (aF d s)) →
+      ∃ bF : (Γ ⧸ N.toSubgroup) → T → H, ∀ i ∈ J, ∀ d,
+        pathProduct (fun s => QuotientGroup.mk' N.toSubgroup (v s))
+          (mergeLabels aF bF) (w i) d ∈ K i) :
+    ∃ b : Γ → T → H, ∀ i, pathProduct v (mergeLabels a b) (w i) (g i) ∈ K i := by
+  classical
+  let C : I → Set (Γ → T → H) := fun i =>
+    {b | pathProduct v (mergeLabels a b) (w i) (g i) ∈ K i}
+  have hc : ∀ i, IsClosed (C i) := fun i =>
+    (hK i).preimage ((pathProduct_continuous v (w i) (g i)).comp (mergeLabels_continuous a))
+  have hf : ∀ J : Finset I, (⋂ i ∈ J, C i).Nonempty := by
+    intro J
+    choose P hP using (fun i : I => pathProduct_finite_locations (H := H) v (w i) (g i))
+    let V := J.biUnion P
+    obtain ⟨N, hN⟩ := finite_set_quotient_injective V
+    let q := QuotientGroup.mk' N.toSubgroup
+    obtain ⟨aF, haF, heF⟩ := copy_labels_on_finite_set q V hN a Allowed h1 ha
+    obtain ⟨bF, hbF⟩ := hsolve N J aF haF
+    refine ⟨fun p t => bF (q p) t, ?_⟩
+    simp only [Set.mem_iInter]
+    intro i hi
+    change pathProduct v (mergeLabels a (fun p t => bF (q p) t)) (w i) (g i) ∈ K i
+    have he : pathProduct v (mergeLabels a (fun p t => bF (q p) t)) (w i) (g i) =
+        pathProduct v (fun p s => mergeLabels aF bF (q p) s) (w i) (g i) := by
+      apply hP i
+      intro p hp s
+      cases s with
+      | inl s => exact (heF p (Finset.mem_biUnion.mpr ⟨i, hi, hp⟩) s).symm
+      | inr t => rfl
+    rw [he, pathProduct_pullback]
+    exact hbF i hi (q (g i))
+  obtain ⟨b, hb⟩ := CompactSpace.iInter_nonempty hc hf
+  exact ⟨b, fun i => Set.mem_iInter.mp hb i⟩
+end CompactLabeling
+end EilenbergGanea
+
+open scoped Quaternion
+namespace EilenbergGanea
+
+theorem su2_norm (a : SU2) : ‖(a : ℍ)‖ = 1 := by
+  have h : star (a : ℍ) * (a : ℍ) = 1 := a.property.1
+  have hn := congrArg norm h
+  simp only [norm_mul, norm_star, norm_one] at hn
+  nlinarith [norm_nonneg (a : ℍ)]
+
+/-- The two elementary estimates needed for the presentation argument, with
+constants independent of the Cayley graph or any finite quotient. -/
+theorem unit_product_linearization (l : List ℍ) (h : ℝ) (hh : 0 ≤ h)
+    (hu : ∀ a ∈ l, ‖a‖ = 1) (hb : ∀ a ∈ l, ‖a - 1‖ ≤ h) :
+    ‖l.prod - 1‖ ≤ l.length * h ∧
+    ‖l.prod - 1 - (l.map (fun a => a - 1)).sum‖ ≤ (l.length : ℝ)^2 * h^2 := by
+  induction l with
+  | nil => simp
+  | cons a l ih =>
+      obtain ⟨hp, he⟩ := ih (fun b hm => hu b (by simp [hm])) (fun b hm => hb b (by simp [hm]))
+      have hna := hu a (by simp)
+      have hba := hb a (by simp)
+      have hid : a * l.prod - 1 = a * (l.prod - 1) + (a - 1) := by noncomm_ring
+      have hie : a * l.prod - 1 - ((a - 1) + (l.map (fun b => b - 1)).sum) =
+          (a - 1) * (l.prod - 1) + (l.prod - 1 - (l.map (fun b => b - 1)).sum) := by noncomm_ring
+      constructor
+      · simp only [List.prod_cons, List.length_cons, Nat.cast_add, Nat.cast_one]
+        rw [hid]
+        calc
+          _ ≤ ‖a * (l.prod - 1)‖ + ‖a - 1‖ := norm_add_le _ _
+          _ = ‖l.prod - 1‖ + ‖a - 1‖ := by rw [norm_mul, hna, one_mul]
+          _ ≤ (l.length + 1) * h := by nlinarith
+      · simp only [List.prod_cons, List.map_cons, List.sum_cons, List.length_cons, Nat.cast_add, Nat.cast_one]
+        rw [hie]
+        have hn : 0 ≤ (l.length : ℝ) := Nat.cast_nonneg _
+        calc
+          _ ≤ ‖(a - 1) * (l.prod - 1)‖ + ‖l.prod - 1 - (l.map (fun b => b - 1)).sum‖ := norm_add_le _ _
+          _ = ‖a - 1‖ * ‖l.prod - 1‖ + ‖l.prod - 1 - (l.map (fun b => b - 1)).sum‖ := by rw [norm_mul]
+          _ ≤ h * (l.length * h) + (l.length : ℝ)^2 * h^2 :=
+            add_le_add (mul_le_mul hba hp (norm_nonneg _) hh) he
+          _ ≤ (l.length + 1)^2 * h^2 := by nlinarith [sq_nonneg h]
+
+ theorem quaternion_commutator_bound (a b : ℍ) :
+    ‖a * b - b * a‖ ≤ 2 * ‖a - 1‖ * ‖b - 1‖ := by
+  have he : a * b - b * a = (a - 1) * (b - 1) - (b - 1) * (a - 1) := by noncomm_ring
+  rw [he]
+  calc
+    _ ≤ ‖(a - 1) * (b - 1)‖ + ‖(b - 1) * (a - 1)‖ := norm_sub_le _ _
+    _ = _ := by rw [norm_mul, norm_mul]; ring
+
+ theorem su2_conjugation_error (a b : SU2) :
+    ‖((a * b * a⁻¹ : SU2) : ℍ) - (b : ℍ)‖ ≤ 2 * ‖(a : ℍ) - 1‖ * ‖(b : ℍ) - 1‖ := by
+  have he : ((a * b * a⁻¹ : SU2) : ℍ) - (b : ℍ) =
+      ((a : ℍ) * (b : ℍ) - (b : ℍ) * (a : ℍ)) * ((a⁻¹ : SU2) : ℍ) := by
+    change (a : ℍ) * (b : ℍ) * star (a : ℍ) - (b : ℍ) = _
+    change _ = ((a : ℍ) * (b : ℍ) - (b : ℍ) * (a : ℍ)) * star (a : ℍ)
+    rw [sub_mul, mul_assoc (b : ℍ), a.property.2, mul_one]
+  rw [he, norm_mul, su2_norm, mul_one]
+  exact quaternion_commutator_bound _ _
+
+ theorem su2_inverse_distance (b : SU2) : ‖((b⁻¹ : SU2) : ℍ) - 1‖ = ‖(b : ℍ) - 1‖ := by
+  change ‖star (b : ℍ) - 1‖ = _
+  calc
+    _ = ‖star ((b : ℍ) - 1)‖ := by rw [star_sub, star_one]
+    _ = _ := norm_star _
+
+ theorem su2_inverse_linearization (b : SU2) :
+    ‖((b⁻¹ : SU2) : ℍ) - 1 + ((b : ℍ) - 1)‖ ≤ ‖(b : ℍ) - 1‖^2 := by
+  have he : ((b⁻¹ : SU2) : ℍ) - 1 + ((b : ℍ) - 1) =
+      -((((b⁻¹ : SU2) : ℍ) - 1) * ((b : ℍ) - 1)) := by
+    have hb : ((b⁻¹ : SU2) : ℍ) * (b : ℍ) = 1 := b.property.1
+    noncomm_ring [hb]
+  rw [he, norm_neg, norm_mul, su2_inverse_distance, pow_two]
+
+/-- Replacing each displacement by its linear model introduces only the sum
+of the errors. Combined with exact signed counts, this avoids any choice of
+an ordering of the relator indices (which can be uncountable). -/
+theorem unit_product_with_linear_models {I : Type*} (l : List I)
+    (f : I → SU2) (v : I → ℍ) (h D : ℝ) (hh : 0 ≤ h)
+    (hb : ∀ i ∈ l, ‖(f i : ℍ) - 1‖ ≤ h)
+    (he : ∀ i ∈ l, ‖(f i : ℍ) - 1 - v i‖ ≤ D) :
+    ‖((l.map (fun i => (f i : ℍ))).prod - 1) - (l.map v).sum‖ ≤
+      (l.length : ℝ)^2 * h^2 + l.length * D := by
+  have hp := (unit_product_linearization (l.map (fun i => (f i : ℍ))) h hh
+    (by intro a ha; obtain ⟨i, _, rfl⟩ := List.mem_map.mp ha; exact su2_norm _)
+    (by intro a ha; obtain ⟨i, hi, rfl⟩ := List.mem_map.mp ha; exact hb i hi)).2
+  simp only [List.length_map, List.map_map, Function.comp_def] at hp
+  have hs : ∀ l : List I, (∀ i ∈ l, ‖(f i : ℍ) - 1 - v i‖ ≤ D) →
+      ‖(l.map (fun i => (f i : ℍ) - 1)).sum - (l.map v).sum‖ ≤ l.length * D := by
+    intro l he
+    induction l with
+    | nil => simp
+    | cons i l ih =>
+        have hi := he i (by simp)
+        have hrest := ih (fun j hj => he j (by simp [hj]))
+        simp only [List.map_cons, List.sum_cons, List.length_cons, Nat.cast_add, Nat.cast_one]
+        have heq : (f i : ℍ) - 1 + (l.map (fun j => (f j : ℍ) - 1)).sum -
+            (v i + (l.map v).sum) = ((f i : ℍ) - 1 - v i) +
+              ((l.map (fun j => (f j : ℍ) - 1)).sum - (l.map v).sum) := by abel
+        rw [heq]
+        exact (norm_add_le _ _).trans (by linarith)
+
+  calc
+    _ ≤ ‖((l.map (fun i => (f i : ℍ))).prod - 1) - (l.map (fun i => (f i : ℍ) - 1)).sum‖ +
+      ‖(l.map (fun i => (f i : ℍ) - 1)).sum - (l.map v).sum‖ := norm_sub_le_norm_sub_add_norm_sub _ _ _
+    _ ≤ _ := add_le_add hp (hs l he)
+end EilenbergGanea
+
+namespace EilenbergGanea
+open scoped Quaternion
+
+theorem su2_conjugation_distance (a b : SU2) :
+    ‖((a * b * a⁻¹ : SU2) : ℍ) - 1‖ = ‖(b : ℍ) - 1‖ := by
+  have he : ((a * b * a⁻¹ : SU2) : ℍ) - 1 =
+      (a : ℍ) * ((b : ℍ) - 1) * star (a : ℍ) := by
+    change (a : ℍ) * (b : ℍ) * star (a : ℍ) - 1 = _
+    rw [mul_sub, sub_mul, mul_one, a.property.2]
+  rw [he, norm_mul, norm_mul, norm_star, su2_norm, one_mul, mul_one]
+
+def signedUnit (b : SU2) (ε : Bool) : SU2 := if ε then b else b⁻¹
+def signedDisplacement (b : SU2) (ε : Bool) : ℍ := if ε then (b : ℍ) - 1 else -((b : ℍ) - 1)
+
+ theorem signedUnit_distance (b : SU2) (ε : Bool) :
+    ‖(signedUnit b ε : ℍ) - 1‖ = ‖(b : ℍ) - 1‖ := by
+  cases ε <;> simp [signedUnit, su2_inverse_distance]
+
+ theorem signedUnit_error (b : SU2) (ε : Bool) :
+    ‖(signedUnit b ε : ℍ) - 1 - signedDisplacement b ε‖ ≤ ‖(b : ℍ) - 1‖^2 := by
+  cases ε with
+  | false =>
+      change ‖((b⁻¹ : SU2) : ℍ) - 1 - (-((b : ℍ) - 1))‖ ≤ _
+      simpa only [sub_neg_eq_add] using su2_inverse_linearization b
+  | true => simp [signedUnit, signedDisplacement]
+
+ theorem conjugated_signedUnit_error (a b : SU2) (ε : Bool) (A h : ℝ)
+    (hA : 0 ≤ A) (hh : 0 ≤ h) (ha : ‖(a : ℍ) - 1‖ ≤ A) (hb : ‖(b : ℍ) - 1‖ ≤ h) :
+    ‖((a * signedUnit b ε * a⁻¹ : SU2) : ℍ) - 1 - signedDisplacement b ε‖ ≤
+      2 * A * h + h^2 := by
+  have he : ((a * signedUnit b ε * a⁻¹ : SU2) : ℍ) - 1 - signedDisplacement b ε =
+      (((a * signedUnit b ε * a⁻¹ : SU2) : ℍ) - (signedUnit b ε : ℍ)) +
+      ((signedUnit b ε : ℍ) - 1 - signedDisplacement b ε) := by abel
+  rw [he]
+  have hsq : ‖(b : ℍ) - 1‖^2 ≤ h^2 := by nlinarith [norm_nonneg ((b : ℍ) - 1)]
+  calc
+    _ ≤ ‖((a * signedUnit b ε * a⁻¹ : SU2) : ℍ) - (signedUnit b ε : ℍ)‖ +
+      ‖(signedUnit b ε : ℍ) - 1 - signedDisplacement b ε‖ := norm_add_le _ _
+    _ ≤ 2 * ‖(a : ℍ) - 1‖ * ‖(b : ℍ) - 1‖ + ‖(b : ℍ) - 1‖^2 := by
+      simpa only [signedUnit_distance] using add_le_add (su2_conjugation_error a (signedUnit b ε)) (signedUnit_error b ε)
+    _ ≤ _ := add_le_add (mul_le_mul (mul_le_mul_of_nonneg_left ha (by positivity)) hb
+      (norm_nonneg _) (mul_nonneg (by positivity) hA)) hsq
+
+abbrev RelatorFactor (I : Type*) := SU2 × I × Bool
+
+def relatorFactorValue {I : Type*} (x : I → SU2) (r : RelatorFactor I) : SU2 :=
+  r.1 * signedUnit (x r.2.1) r.2.2 * r.1⁻¹
+
+def relatorSignedCounts {I : Type*} (l : List (RelatorFactor I)) : I →₀ ℤ :=
+  (l.map (fun r => Finsupp.single r.2.1 (if r.2.2 then 1 else -1))).sum
+
+ theorem relator_linear_models {I : Type*} (l : List (RelatorFactor I)) (x : I → SU2) :
+    (l.map (fun r => signedDisplacement (x r.2.1) r.2.2)).sum =
+      Finsupp.linearCombination ℤ (fun i => (x i : ℍ) - 1) (relatorSignedCounts l) := by
+  induction l with
+  | nil => simp [relatorSignedCounts]
+  | cons r l ih =>
+      simp only [List.map_cons, List.sum_cons, relatorSignedCounts, map_add, Finsupp.linearCombination_single]
+      rw [← show Finsupp.linearCombination ℤ (fun i => (x i : ℍ) - 1)
+          ((l.map (fun r => Finsupp.single r.2.1 (if r.2.2 then 1 else -1))).sum) =
+          (l.map (fun r => signedDisplacement (x r.2.1) r.2.2)).sum from ih.symm]
+      congr 2
+      cases r.2.2 <;> simp [signedDisplacement]
+
+/-- Exact translated counts cancel the full linear term; the remaining bound
+is quadratic. The index set is arbitrary, not necessarily countable. -/
+theorem relator_product_quadratic {I : Type*} (l : List (RelatorFactor I)) (x : I → SU2)
+    (i₀ : I) (hc : relatorSignedCounts l = Finsupp.single i₀ 1)
+    (h ε K : ℝ) (hh : 0 ≤ h) (hε : 0 ≤ ε) (hK : 0 ≤ K)
+    (hb : ∀ r ∈ l, ‖(x r.2.1 : ℍ) - 1‖ ≤ h)
+    (ha : ∀ r ∈ l, ‖(r.1 : ℍ) - 1‖ ≤ K * (ε + h)) :
+    ‖(l.map (fun r => (relatorFactorValue x r : ℍ))).prod - (x i₀ : ℍ)‖ ≤
+      ((l.length : ℝ)^2 + l.length + 2 * l.length * K) * (ε + h) * h := by
+  have hp := unit_product_with_linear_models l (relatorFactorValue x)
+    (fun r => signedDisplacement (x r.2.1) r.2.2) h
+    (2 * (K * (ε + h)) * h + h^2) hh
+    (fun r hr => by rw [relatorFactorValue, su2_conjugation_distance, signedUnit_distance]; exact hb r hr)
+    (fun r hr => conjugated_signedUnit_error r.1 (x r.2.1) r.2.2 _ h
+      (mul_nonneg hK (add_nonneg hε hh)) hh (ha r hr) (hb r hr))
+  rw [relator_linear_models, hc, Finsupp.linearCombination_single, one_smul] at hp
+  have he : (l.map (fun r => (relatorFactorValue x r : ℍ))).prod - 1 - ((x i₀ : ℍ) - 1) =
+      (l.map (fun r => (relatorFactorValue x r : ℍ))).prod - (x i₀ : ℍ) := by abel
+  rw [he] at hp
+  apply hp.trans
+  have hlen : 0 ≤ (l.length : ℝ) := Nat.cast_nonneg _
+  have hnh : 0 ≤ ((l.length : ℝ)^2 + l.length) * ε * h :=
+    mul_nonneg (mul_nonneg (add_nonneg (sq_nonneg _) hlen) hε) hh
+  nlinarith
+end EilenbergGanea
+
+namespace EilenbergGanea
+
+/-- A uniform size for an arbitrary family, normalized to zero for an empty
+index set. This is a real supremum, not an attained maximum. -/
+def familySize {I : Type*} (f : I → ℝ) : ℝ := sSup (Set.range f ∪ {0})
+
+ theorem familySize_nonneg {I : Type*} (f : I → ℝ) (hf : BddAbove (Set.range f)) :
+    0 ≤ familySize f := by
+  apply le_csSup (hf.union bddAbove_singleton)
+  exact Or.inr (Set.mem_singleton _)
+
+ theorem le_familySize {I : Type*} (f : I → ℝ) (hf : BddAbove (Set.range f)) (i : I) :
+    f i ≤ familySize f := le_csSup (hf.union bddAbove_singleton) (Or.inl (Set.mem_range_self i))
+
+ theorem familySize_le {I : Type*} (f : I → ℝ) {c : ℝ} (hc : 0 ≤ c) (hf : ∀ i, f i ≤ c) :
+    familySize f ≤ c := by
+  apply csSup_le (Set.union_nonempty.mpr (Or.inr (Set.singleton_nonempty 0)))
+  rintro x (⟨i, rfl⟩ | rfl)
+  · exact hf i
+  · exact hc
+
+/-- The source no-threshold-crossing argument does not assume attainment of
+its supremum. -/
+ theorem familySize_ne_threshold {I : Type*} (f : I → ℝ)
+    (hf : BddAbove (Set.range f)) (C δ η ε : ℝ)
+    (hδ : 0 < δ) (hC : 0 ≤ C) (_hη : 0 ≤ η) (hε : 0 ≤ ε) (hεη : ε ≤ η)
+    (hc₁ : C * (δ + η) < 1/2) (hc₂ : C * η < δ/2)
+    (hbound : ∀ i, f i ≤ C * ε + C * (familySize f + ε) * familySize f) :
+    familySize f ≠ δ := by
+  intro he
+  have hs : familySize f ≤ C * ε + C * (familySize f + ε) * familySize f :=
+    familySize_le f (by positivity [familySize_nonneg f hf]) hbound
+  rw [he] at hs
+  have hce : C * ε ≤ C * η := mul_le_mul_of_nonneg_left hεη hC
+  have hfactor : C * (δ + ε) ≤ C * (δ + η) := mul_le_mul_of_nonneg_left (by linarith) hC
+  nlinarith [mul_lt_mul_of_pos_right hc₁ hδ, mul_le_mul_of_nonneg_right hfactor hδ.le]
+
+/-- Uniform contraction of all residual errors. This handles even an
+uncountable family and does not replace the supremum by a maximum. -/
+ theorem familySize_contraction {I : Type*} (f : I → ℝ)
+    (hf : ∀ i, 0 ≤ f i) (δ C η : ℝ) (hδ : 0 ≤ δ) (hC : 0 ≤ C) (hη : 0 ≤ η)
+    (hb : ∀ i, f i ≤ δ) (hc : C * (δ + η) < 1)
+    (he : ∀ i, f i ≤ C * (familySize f + η) * familySize f) : ∀ i, f i = 0 := by
+  have hbounded : BddAbove (Set.range f) := ⟨δ, by rintro _ ⟨i, rfl⟩; exact hb i⟩
+  have h0 := familySize_nonneg f hbounded
+  have hmax := familySize_le f hδ hb
+  have hs := familySize_le f (c := C * (familySize f + η) * familySize f) (by positivity) he
+  have hmul : C * (familySize f + η) ≤ C * (δ + η) :=
+    mul_le_mul_of_nonneg_left (by linarith) hC
+  have heq : familySize f = 0 := by
+    by_contra hn
+    have hp : 0 < familySize f := lt_of_le_of_ne h0 (Ne.symm hn)
+    have ht := mul_lt_mul_of_pos_right (lt_of_le_of_lt hmul hc) hp
+    nlinarith
+  intro i
+  exact le_antisymm (by simpa only [heq] using le_familySize f hbounded i) (hf i)
+
+ theorem small_thresholds_exist (C : ℝ) (hC : 1 ≤ C) :
+    ∃ δ η : ℝ, 0 < δ ∧ δ < 1 ∧ 0 < η ∧ η < 1 ∧
+      C * (δ + η) < 1/2 ∧ C * η < δ/2 := by
+  have hc : 0 < C := by linarith
+  let δ := 1 / (8*C)
+  let η := δ / (8*C)
+  have hd : 0 < δ := by dsimp [δ]; positivity
+  have hd1 : δ < 1 := by dsimp [δ]; apply (div_lt_iff₀ (by positivity)).mpr; linarith
+  have he : 0 < η := by dsimp [η]; positivity
+  have heδ : η < δ := by dsimp [η]; apply (div_lt_iff₀ (by positivity)).mpr; nlinarith
+  have hcd : C * δ = 1/8 := by dsimp [δ]; field_simp
+  have hce : C * η = δ/8 := by dsimp [η]; field_simp
+  refine ⟨δ, η, hd, hd1, he, heδ.trans hd1, ?_, ?_⟩
+  · rw [mul_add, hcd, hce]; linarith
+  · rw [hce]; linarith
+end EilenbergGanea
+
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/UnitaryWords.lean

@@ -1,890 +1,890 @@
-import OAI.Topology.EilenbergGanea.Regularization
-import Mathlib.Analysis.Quaternion
-import Mathlib.Analysis.Complex.Circle
-import Mathlib.Analysis.Complex.Polynomial.Basic
-import Mathlib.Analysis.Calculus.Deriv.ZPow
-import Mathlib.GroupTheory.FreeGroup.Basic
-import Mathlib.Analysis.SpecialFunctions.SmoothTransition
-
-namespace OAI
-
-noncomputable section
-
-open Set Filter Topology
-open scoped Quaternion ContDiff
-
-open scoped Quaternion
-namespace EilenbergGanea
-abbrev SU2 := unitary ℍ
-
-theorem quaternion_mem_unitary_iff (q : ℍ) : q ∈ unitary ℍ ↔ ‖q‖ = 1 := by
-  rw [Unitary.mem_iff, Quaternion.star_mul_self, Quaternion.self_mul_star, and_self]
-  constructor
-  · intro h
-    have he : Quaternion.normSq q = 1 := Quaternion.coe_inj.mp h
-    rw [Quaternion.normSq_eq_norm_mul_self] at he
-    nlinarith [norm_nonneg q]
-  · intro h
-    simp [Quaternion.normSq_eq_norm_mul_self, h]
-
-instance su2_compactSpace : CompactSpace SU2 := by
-  apply isCompact_iff_compactSpace.mp
-  have hs : (unitary ℍ : Set ℍ) = Metric.sphere 0 1 := by
-    ext q
-    simp [quaternion_mem_unitary_iff]
-  rw [hs]
-  exact isCompact_sphere 0 1
-
-def circleQuaternion : Circle →* SU2 where
-  toFun z := ⟨Quaternion.ofComplex z, by
-    rw [Unitary.mem_iff, Quaternion.star_mul_self, Quaternion.self_mul_star, and_self]
-    have hz := Circle.normSq_coe z
-    have he : Quaternion.normSq (Quaternion.ofComplex z) = 1 := by
-      simpa [Quaternion.normSq_def', Complex.normSq_apply, sq] using hz
-    rw [he]; rfl⟩
-  map_one' := Subtype.ext (map_one Quaternion.ofComplex)
-  map_mul' z w := Subtype.ext (map_mul Quaternion.ofComplex (z : ℂ) (w : ℂ))
-
-@[simp] theorem circleQuaternion_coe (z : Circle) :
-    ((circleQuaternion z : SU2) : ℍ) = (z : ℂ) := rfl
-
-theorem circleQuaternion_injective : Function.Injective circleQuaternion := by
-  intro z w h
-  apply Subtype.ext
-  apply Complex.ext
-  · exact congrArg (fun q : ℍ => q.re) (congrArg Subtype.val h)
-  · exact congrArg (fun q : ℍ => q.imI) (congrArg Subtype.val h)
-
-theorem circleQuaternion_continuous : Continuous circleQuaternion := by
-  apply Continuous.subtype_mk
-  exact Quaternion.ofComplex.toLinearMap.continuous_of_finiteDimensional.comp continuous_subtype_val
-
-def wordMap {m : ℕ} (w : Fin m → FreeGroup (Fin m)) (x : Fin m → SU2) : Fin m → SU2 :=
-  fun i => FreeGroup.lift x (w i)
-
-def exponentMatrix {m : ℕ} (w : Fin m → FreeGroup (Fin m)) : Matrix (Fin m) (Fin m) ℤ :=
-  fun i j => Multiplicative.toAdd
-    (FreeGroup.lift (fun k => Multiplicative.ofAdd (if j = k then (1 : ℤ) else 0)) (w i))
-
-open Set Filter Topology
-open scoped ContDiff
-
-/-- Radial normalization is only used on nonzero quaternions. The harmless
-value at zero makes it a total function without changing that domain. -/
-def normalizeQuaternion (q : ℍ) : SU2 := by
-  classical
-  exact if h : q = 0 then 1 else
-    ⟨‖q‖⁻¹ • q, (quaternion_mem_unitary_iff _).mpr (by
-      simp only [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm]
-      exact inv_mul_cancel₀ (norm_ne_zero_iff.mpr h))⟩
-
-theorem normalizeQuaternion_coe_of_ne_zero {q : ℍ} (hq : q ≠ 0) :
-    (normalizeQuaternion q : ℍ) = ‖q‖⁻¹ • q := by simp [normalizeQuaternion, hq]
-
-theorem normalizeQuaternion_coe_contDiffAt {q : ℍ} (hq : q ≠ 0) :
-    ContDiffAt ℝ ∞ (fun z => (normalizeQuaternion z : ℍ)) q := by
-  have h : ContDiffAt ℝ ∞ (fun z : ℍ => ‖z‖⁻¹ • z) q :=
-    ((contDiffAt_norm ℝ hq).inv (norm_ne_zero_iff.mpr hq)).smul contDiffAt_id
-  apply h.congr_of_eventuallyEq
-  filter_upwards [eventually_ne_nhds hq] with z hz
-  exact normalizeQuaternion_coe_of_ne_zero hz
-
-theorem normalizeQuaternion_continuousAt {q : ℍ} (hq : q ≠ 0) :
-    ContinuousAt normalizeQuaternion q := by
-  exact tendsto_subtype_rng.mpr (normalizeQuaternion_coe_contDiffAt hq).continuousAt
-
-@[simp] theorem normalizeQuaternion_unitary (q : SU2) : normalizeQuaternion (q : ℍ) = q := by
-  have hqnorm : ‖(q : ℍ)‖ = 1 := (quaternion_mem_unitary_iff _).mp q.property
-  have hq : (q : ℍ) ≠ 0 := by intro he; simp [he] at hqnorm
-  apply Subtype.ext
-  simp [normalizeQuaternion_coe_of_ne_zero hq, hqnorm]
-
-/-- Smoothness of coefficient-free word evaluation follows directly by
-induction, using quaternion conjugation for unitary inverses. -/
-theorem freeGroup_lift_coe_contDiffAt {D ι : Type*}
-    [NormedAddCommGroup D] [NormedSpace ℝ D] (w : FreeGroup ι)
-    (a : D) (x : D → ι → SU2)
-    (hx : ∀ i, ContDiffAt ℝ ∞ (fun z => (x z i : ℍ)) a) :
-    ContDiffAt ℝ ∞ (fun z => ((FreeGroup.lift (x z) w : SU2) : ℍ)) a := by
-  induction w using FreeGroup.induction_on with
-  | one => simpa using (contDiffAt_const : ContDiffAt ℝ ∞ (fun _ : D => (1 : ℍ)) a)
-  | of i => simpa using hx i
-  | inv_of i hi =>
-      simp only [map_inv]
-      change ContDiffAt ℝ ∞ (fun z => star ((FreeGroup.lift (x z) (FreeGroup.of i) : SU2) : ℍ)) a
-      let L : ℍ →ₗ[ℝ] ℍ :=
-        { toFun := star
-          map_add' := star_add
-          map_smul' := Quaternion.star_smul }
-      exact L.toContinuousLinearMap.contDiff.contDiffAt.comp a hi
-  | mul v w hv hw => simpa only [map_mul, Submonoid.coe_mul] using hv.mul hw
-
-def radialWordMap {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (x : Fin m → ℍ) : Fin m → ℍ :=
-  fun i => ‖x i‖ • ((wordMap w (fun j => normalizeQuaternion (x j)) i : SU2) : ℍ)
-
-@[simp] theorem radialWordMap_norm {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (x : Fin m → ℍ) (i : Fin m) : ‖radialWordMap w x i‖ = ‖x i‖ := by
-  have h : ‖((wordMap w (fun j => normalizeQuaternion (x j)) i : SU2) : ℍ)‖ = 1 :=
-    (quaternion_mem_unitary_iff _).mp (wordMap w (fun j => normalizeQuaternion (x j)) i).property
-  simp [radialWordMap, norm_smul, h]
-
-theorem radialWordMap_contDiffAt {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (x : Fin m → ℍ) (hx : ∀ i, x i ≠ 0) : ContDiffAt ℝ ∞ (radialWordMap w) x := by
-  apply contDiffAt_pi.mpr
-  intro i
-  exact ((contDiffAt_apply ℝ ℍ i x).norm ℝ (hx i)).smul
-    (freeGroup_lift_coe_contDiffAt (w i) x (fun z j => normalizeQuaternion (z j))
-      (fun j => (normalizeQuaternion_coe_contDiffAt (hx j)).comp (f := fun z : Fin m → ℍ => z j) x (contDiffAt_apply ℝ ℍ j x)))
-
-theorem radialWordMap_eq_one_iff {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (x : Fin m → ℍ) : radialWordMap w x = 1 ↔
-      (∀ i, ‖x i‖ = 1) ∧ wordMap w (fun j => normalizeQuaternion (x j)) = 1 := by
-  constructor
-  · intro h
-    have hn (i : Fin m) : ‖x i‖ = 1 := by
-      have he := congrArg (fun y : Fin m → ℍ => ‖y i‖) h
-      simpa only [radialWordMap_norm, Pi.one_apply, norm_one] using he
-    refine ⟨hn, ?_⟩
-    funext i
-    apply Subtype.ext
-    have he := congrFun h i
-    simpa [radialWordMap, hn] using he
-  · rintro ⟨hn, hw⟩
-    funext i
-    simp [radialWordMap, hn, hw]
-
-
-/-- A conjugation-invariant smooth cutoff, zero near coordinate hyperplanes
-and one near the unit quaternion locus. -/
-def radialCutoff {m : ℕ} (x : Fin m → ℍ) : ℝ :=
-  ∏ i, Real.smoothTransition (4 * ‖x i‖ ^ 2 - 1)
-
-def smoothRadialWordMap {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (x : Fin m → ℍ) : Fin m → ℍ := radialCutoff x • radialWordMap w x
-
-theorem radialCutoff_contDiff {m : ℕ} : ContDiff ℝ ∞ (@radialCutoff m) := by
-  apply contDiff_prod
-  intro i _
-  have hn : ContDiff ℝ ∞ (fun x : Fin m → ℍ => ‖x i‖ ^ 2) :=
-    (contDiff_apply ℝ ℍ i).norm_sq ℝ
-  exact Real.smoothTransition.contDiff.comp ((contDiff_const.mul hn).sub contDiff_const)
-
-theorem radialCutoff_nonneg {m : ℕ} (x : Fin m → ℍ) : 0 ≤ radialCutoff x :=
-  Finset.prod_nonneg (fun _ _ => Real.smoothTransition.nonneg _)
-
-theorem radialCutoff_le_one {m : ℕ} (x : Fin m → ℍ) : radialCutoff x ≤ 1 :=
-  Finset.prod_le_one₀ (fun _ _ => Real.smoothTransition.nonneg _)
-    (fun _ _ => Real.smoothTransition.le_one _)
-
-theorem radialCutoff_eq_one {m : ℕ} (x : Fin m → ℍ) (hx : ∀ i, 1 ≤ ‖x i‖) :
-    radialCutoff x = 1 := by
-  apply Finset.prod_eq_one
-  intro i _
-  apply Real.smoothTransition.one_of_one_le
-  nlinarith [hx i]
-
-theorem radialCutoff_eventually_zero {m : ℕ} (x : Fin m → ℍ) (i : Fin m)
-    (hi : x i = 0) : radialCutoff =ᶠ[𝓝 x] (fun _ => 0) := by
-  have h : ∀ᶠ y in 𝓝 x, ‖y i‖ < (1 / 2 : ℝ) :=
-    ((continuous_apply i).norm.continuousAt.eventually_lt_const (by simp [hi]))
-  filter_upwards [h] with y hy
-  apply Finset.prod_eq_zero (Finset.mem_univ i)
-  apply Real.smoothTransition.zero_of_nonpos
-  nlinarith [norm_nonneg (y i)]
-
-theorem radialCutoff_eventually_one {m : ℕ} (x : Fin m → ℍ)
-    (hx : ∀ i, ‖x i‖ = 1) : radialCutoff =ᶠ[𝓝 x] (fun _ => 1) := by
-  have h (i : Fin m) : ∀ᶠ y in 𝓝 x, (3 / 4 : ℝ) < ‖y i‖ :=
-    ((continuous_apply i).norm.continuousAt.eventually_const_lt (by change (3 / 4 : ℝ) < ‖x i‖; rw [hx i]; norm_num))
-  filter_upwards [Filter.eventually_all.mpr h] with y hy
-  apply Finset.prod_eq_one
-  intro i _
-  apply Real.smoothTransition.one_of_one_le
-  nlinarith [hy i]
-
-theorem smoothRadialWordMap_contDiff {m : ℕ} (w : Fin m → FreeGroup (Fin m)) :
-    ContDiff ℝ ∞ (smoothRadialWordMap w) := by
-  rw [contDiff_iff_contDiffAt]
-  intro x
-  by_cases hx : ∀ i, x i ≠ 0
-  · exact radialCutoff_contDiff.contDiffAt.smul (radialWordMap_contDiffAt w x hx)
-  · push Not at hx
-    obtain ⟨i, hi⟩ := hx
-    apply (contDiffAt_const : ContDiffAt ℝ ∞ (fun _ : Fin m → ℍ => (0 : Fin m → ℍ)) x).congr_of_eventuallyEq
-    filter_upwards [radialCutoff_eventually_zero x i hi] with y hy
-    simp [smoothRadialWordMap, hy]
-
-theorem smoothRadialWordMap_eq_one_iff {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (x : Fin m → ℍ) : smoothRadialWordMap w x = 1 ↔ radialWordMap w x = 1 := by
-  constructor
-  · intro h
-    have hn (i : Fin m) : radialCutoff x * ‖x i‖ = 1 := by
-      have he := congrArg (fun y : Fin m → ℍ => ‖y i‖) h
-      simpa only [smoothRadialWordMap, Pi.smul_apply, norm_smul, Real.norm_eq_abs,
-        abs_of_nonneg (radialCutoff_nonneg x), radialWordMap_norm, Pi.one_apply, norm_one] using he
-    have hlarge (i : Fin m) : 1 ≤ ‖x i‖ := by
-      have hle := mul_le_mul_of_nonneg_right (radialCutoff_le_one x) (norm_nonneg (x i))
-      rw [hn i, one_mul] at hle
-      exact hle
-    simpa only [smoothRadialWordMap, radialCutoff_eq_one x hlarge, one_smul] using h
-  · intro h
-    have hn := (radialWordMap_eq_one_iff w x).mp h
-    simp only [smoothRadialWordMap, radialCutoff_eq_one x (fun i => (hn.1 i).ge), one_smul, h]
-
-theorem smoothRadialWordMap_eventuallyEq {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (x : Fin m → ℍ) (hx : ∀ i, ‖x i‖ = 1) :
-    smoothRadialWordMap w =ᶠ[𝓝 x] radialWordMap w := by
-  filter_upwards [radialCutoff_eventually_one x hx] with y hy
-  simp [smoothRadialWordMap, hy]
-
-end EilenbergGanea
-
-namespace EilenbergGanea
-open Set Filter Topology
-variable {n : Type*} [Fintype n] [DecidableEq n]
-
-def splitComplex : (n → ℂ) ≃ₗ[ℝ] ((n → ℝ) × (n → ℝ)) where
-  toFun z := (fun i => (z i).re, fun i => (z i).im)
-  invFun p i := (p.1 i : ℂ) + (p.2 i : ℂ) * Complex.I
-  left_inv z := by funext i; exact Complex.re_add_im (z i)
-  right_inv p := by ext i <;> simp
-  map_add' a b := by ext i <;> simp
-  map_smul' a b := by ext i <;> simp
-
-def radialAngularLinear (B : Matrix n n ℝ) : (n → ℂ) →L[ℝ] (n → ℂ) :=
-  ((splitComplex (n := n)).symm.toLinearMap.comp
-    (((LinearMap.id : (n → ℝ) →ₗ[ℝ] (n → ℝ)).prodMap B.toLin').comp
-      (splitComplex (n := n)).toLinearMap)).toContinuousLinearMap
-
-@[simp] theorem radialAngularLinear_apply (B : Matrix n n ℝ) (z : n → ℂ) (i : n) :
-    radialAngularLinear B z i = ((z i).re : ℂ) + (∑ j, B i j * (z j).im : ℝ) * Complex.I := rfl
-
-theorem radialAngularLinear_det (B : Matrix n n ℝ) :
-    (radialAngularLinear B).toLinearMap.det = B.det := by
-  change LinearMap.det ((splitComplex (n := n)).symm.toLinearMap ∘ₗ
-    ((LinearMap.id : (n → ℝ) →ₗ[ℝ] (n → ℝ)).prodMap B.toLin') ∘ₗ
-    (splitComplex (n := n)).toLinearMap) = _
-  have he := LinearMap.det_conj
-    ((LinearMap.id : (n → ℝ) →ₗ[ℝ] (n → ℝ)).prodMap B.toLin')
-    (splitComplex (n := n)).symm
-  rw [LinearEquiv.symm_symm] at he
-  rw [he, LinearMap.det_prodMap, LinearMap.det_id, one_mul, LinearMap.det_toLin']
-
-def complexPolarChart (x z : n → ℂ) : n → ℂ := fun i => x i * Complex.exp (z i)
-
-def complexRadialTorus (B : Matrix n n ℤ) (z : n → ℂ) : n → ℂ :=
-  fun i => ‖z i‖ • ∏ j, (‖z j‖⁻¹ • z j) ^ B i j
-
-theorem normalize_complex_mul_exp (x z : ℂ) (hx : ‖x‖ = 1) :
-    ‖x * Complex.exp z‖⁻¹ • (x * Complex.exp z) =
-      x * Complex.exp ((z.im : ℂ) * Complex.I) := by
-  rw [norm_mul, hx, one_mul, Complex.norm_exp, Complex.real_smul, Complex.ofReal_inv]
-  have he : Complex.exp z = (Real.exp z.re : ℂ) *
-      Complex.exp ((z.im : ℂ) * Complex.I) := by
-    rw [Complex.ofReal_exp, ← Complex.exp_add, Complex.re_add_im]
-  rw [he]
-  have hn : (Real.exp z.re : ℂ) ≠ 0 := by exact_mod_cast Real.exp_ne_zero z.re
-  field_simp
-
- theorem complexRadialTorus_chart (B : Matrix n n ℤ) (x : n → ℂ)
-    (hx : ∀ i, ‖x i‖ = 1) (hroot : ∀ i, ∏ j, x j ^ B i j = 1) :
-    complexRadialTorus B ∘ complexPolarChart x =
-      fun z => fun i => Complex.exp (radialAngularLinear (B.map (Int.castRingHom ℝ)) z i) := by
-  funext z i
-  have hn (j : n) := normalize_complex_mul_exp (x j) (z j) (hx j)
-  simp only [Function.comp_apply, complexRadialTorus, complexPolarChart]
-  simp only [hn]
-  simp only [mul_zpow, Finset.prod_mul_distrib,
-    hroot, one_mul, norm_mul, hx, one_mul, Complex.norm_exp, radialAngularLinear_apply,
-    ← Complex.exp_int_mul]
-  rw [Complex.real_smul, Complex.ofReal_exp, ← Complex.exp_sum, ← Complex.exp_add]
-  congr 1
-  simp only [Matrix.map_apply, Int.coe_castRingHom, Complex.ofReal_sum,
-    Complex.ofReal_mul, Complex.ofReal_intCast]
-  rw [Finset.sum_mul]
-  congr 1
-  apply Finset.sum_congr rfl
-  intro j _
-  ring
-
-
-/-- The real derivative of multiplication by a unit complex tuple has determinant one. -/
-def complexDiagonal (x : n → ℂ) : (n → ℂ) →L[ℝ] (n → ℂ) :=
-  ((Matrix.diagonal x).toLin').toContinuousLinearMap.restrictScalars ℝ
-
-@[simp] theorem complexDiagonal_apply (x z : n → ℂ) (i : n) :
-    complexDiagonal x z i = x i * z i := by simp [complexDiagonal, Matrix.toLin'_apply, Matrix.mulVec, Matrix.diagonal_apply, dotProduct]
-
-theorem complexDiagonal_det (x : n → ℂ) (hx : ∀ i, ‖x i‖ = 1) :
-    (complexDiagonal x).toLinearMap.det = 1 := by
-  change (((Matrix.diagonal x).toLin').restrictScalars ℝ).det = _
-  rw [LinearMap.det_restrictScalars, Algebra.norm_complex_apply, LinearMap.det_toLin',
-    Matrix.det_diagonal]
-  rw [Complex.normSq_eq_norm_sq, norm_prod]
-  simp [hx]
-
-theorem complexPolarChart_hasFDerivAt (x : n → ℂ) :
-    HasFDerivAt (complexPolarChart x) (complexDiagonal x) 0 := by
-  apply hasFDerivAt_pi'.mpr
-  intro i
-  have h := ((ContinuousLinearMap.proj i : (n → ℂ) →L[ℝ] ℂ).hasFDerivAt
-    (x := 0)).cexp.const_mul (x i)
-  change HasFDerivAt (fun z : n → ℂ => x i * Complex.exp (z i)) _ (0 : n → ℂ)
-  have he : (ContinuousLinearMap.proj i : (n → ℂ) →L[ℝ] ℂ).comp (complexDiagonal x) =
-      x i • (ContinuousLinearMap.proj i : (n → ℂ) →L[ℝ] ℂ) := by
-    ext z
-    simp
-  rw [he]
-  simpa using h
-
-theorem exp_radialAngularLinear_hasFDerivAt (B : Matrix n n ℝ) :
-    HasFDerivAt (fun z i => Complex.exp (radialAngularLinear B z i))
-      (radialAngularLinear B) 0 := by
-  apply hasFDerivAt_pi'.mpr
-  intro i
-  simpa using (((ContinuousLinearMap.proj i : (n → ℂ) →L[ℝ] ℂ).comp
-    (radialAngularLinear B)).hasFDerivAt (x := 0)).cexp
-
-theorem complexRadialTorus_differentiableAt (B : Matrix n n ℤ) (x : n → ℂ)
-    (hx : ∀ i, x i ≠ 0) : DifferentiableAt ℝ (complexRadialTorus B) x := by
-  apply differentiableAt_pi.mpr
-  intro i
-  have hn (j : n) : DifferentiableAt ℝ (fun z : n → ℂ => ‖z j‖) x :=
-    (differentiableAt_apply j x).norm ℝ (hx j)
-  apply (hn i).smul
-  have hp (j : n) : DifferentiableAt ℝ (fun z : n → ℂ => (‖z j‖⁻¹ • z j) ^ B i j) x := by
-    have hne : ‖x j‖⁻¹ • x j ≠ 0 :=
-      smul_ne_zero (inv_ne_zero (norm_ne_zero_iff.mpr (hx j))) (hx j)
-    have hz : DifferentiableAt ℝ (fun z : ℂ => z ^ B i j) (‖x j‖⁻¹ • x j) :=
-      ((differentiableAt_zpow (𝕜 := ℂ) (m := B i j)).mpr (Or.inl hne)).restrictScalars ℝ
-    exact hz.comp x (((hn j).inv (norm_ne_zero_iff.mpr (hx j))).smul
-      (differentiableAt_apply j x))
-  exact (HasFDerivAt.finsetProd (fun j _ => (hp j).hasFDerivAt)).differentiableAt
-
-/-- At every torus root, the radial extension has exactly determinant det B.
-The extra radial coordinates contribute the identity, not an additional sign. -/
-theorem complexRadialTorus_det (B : Matrix n n ℤ) (x : n → ℂ)
-    (hx : ∀ i, ‖x i‖ = 1) (hroot : ∀ i, ∏ j, x j ^ B i j = 1) :
-    (fderiv ℝ (complexRadialTorus B) x).det = (B.det : ℝ) := by
-  have hne (i : n) : x i ≠ 0 := by intro h; simpa [h] using hx i
-  have hd := (complexRadialTorus_differentiableAt B x hne).hasFDerivAt
-  have hchart : complexPolarChart x 0 = x := by ext i; simp [complexPolarChart]
-  have hc := (show HasFDerivAt (complexRadialTorus B) (fderiv ℝ (complexRadialTorus B) x)
-      (complexPolarChart x 0) by rw [hchart]; exact hd).comp 0 (complexPolarChart_hasFDerivAt x)
-  have he : (fderiv ℝ (complexRadialTorus B) x).comp (complexDiagonal x) =
-      radialAngularLinear (B.map (Int.castRingHom ℝ)) := by
-    apply hc.unique
-    rw [complexRadialTorus_chart B x hx hroot]
-    exact exp_radialAngularLinear_hasFDerivAt _
-  have hdet := congrArg (fun L : (n → ℂ) →L[ℝ] (n → ℂ) => L.det) he
-  change ((fderiv ℝ (complexRadialTorus B) x).toLinearMap.comp
-    (complexDiagonal x).toLinearMap).det = (radialAngularLinear (B.map (Int.castRingHom ℝ))).toLinearMap.det at hdet
-  rw [LinearMap.det_comp, complexDiagonal_det x hx,
-    mul_one, radialAngularLinear_det] at hdet
-  exact hdet.trans ((Int.castRingHom ℝ).map_det B).symm
-
-end EilenbergGanea
-namespace EilenbergGanea
-section CommutativeWords
-variable {n A : Type*} [Fintype n] [DecidableEq n] [CommGroup A]
-def wordExponents {ι : Type*} [DecidableEq ι] (w : FreeGroup ι) (j : ι) : ℤ :=
-  Multiplicative.toAdd
-    (FreeGroup.lift (fun k => Multiplicative.ofAdd (if j = k then (1 : ℤ) else 0)) w)
-
-@[simp] theorem wordExponents_one {ι : Type*} [DecidableEq ι] (j : ι) : wordExponents (1 : FreeGroup ι) j = 0 := by
-  simp [wordExponents]
-
-@[simp] theorem wordExponents_of {ι : Type*} [DecidableEq ι] (j k : ι) :
-    wordExponents (FreeGroup.of k) j = if j = k then 1 else 0 := by
-  simp [wordExponents]
-
-@[simp] theorem wordExponents_inv {ι : Type*} [DecidableEq ι] (w : FreeGroup ι) (j : ι) :
-    wordExponents w⁻¹ j = -wordExponents w j := by
-  simp [wordExponents]
-
-@[simp] theorem wordExponents_mul {ι : Type*} [DecidableEq ι] (w v : FreeGroup ι) (j : ι) :
-    wordExponents (w * v) j = wordExponents w j + wordExponents v j := by
-  simp [wordExponents]
-
-theorem freeGroup_lift_commutative (x : n → A) (w : FreeGroup n) :
-    FreeGroup.lift x w = ∏ j, x j ^ wordExponents w j := by
-  induction w using FreeGroup.induction_on with
-  | one => simp
-  | of i => simp
-  | inv_of i hi => simp [hi, Finset.prod_inv_distrib]
-  | mul a b ha hb => simp [ha, hb, zpow_add, Finset.prod_mul_distrib]
-
-end CommutativeWords
-end EilenbergGanea
-namespace EilenbergGanea
-open Set Filter Topology
-open scoped Quaternion ContDiff
-
-def quaternionCoordinates : ℍ ≃ₗ[ℝ] ℂ × ℂ where
-  toFun q := (⟨q.re, q.imI⟩, ⟨q.imJ, q.imK⟩)
-  invFun z := ⟨z.1.re, z.1.im, z.2.re, z.2.im⟩
-  left_inv q := by cases q; rfl
-  right_inv z := by rcases z with ⟨⟨a,b⟩,⟨c,d⟩⟩; rfl
-  map_add' q r := by ext <;> rfl
-  map_smul' r q := by apply Prod.ext <;> apply Complex.ext <;> simp
-
-@[simp] theorem quaternionCoordinates_apply (q : ℍ) :
-    quaternionCoordinates q = (⟨q.re, q.imI⟩, ⟨q.imJ, q.imK⟩) := rfl
-
-@[simp] theorem quaternionCoordinates_ofComplex (z : ℂ) :
-    quaternionCoordinates (Quaternion.ofComplex z) = (z, 0) := by
-  ext <;> rfl
-
-@[simp] theorem norm_quaternion_ofComplex (z : ℂ) : ‖Quaternion.ofComplex z‖ = ‖z‖ := by
-  have h : ‖Quaternion.ofComplex z‖ ^ 2 = ‖z‖ ^ 2 := by
-    rw [sq, ← Quaternion.normSq_eq_norm_mul_self, ← Complex.normSq_eq_norm_sq]
-    simp [Quaternion.normSq_def', Complex.normSq_apply, sq]
-  nlinarith [norm_nonneg (Quaternion.ofComplex z), norm_nonneg z]
-
-def quaternionTupleCoordinates {m : ℕ} :
-    (Fin m → ℍ) ≃L[ℝ] ((Fin m → ℂ) × (Fin m → ℂ)) :=
-  (show (Fin m → ℍ) ≃ₗ[ℝ] ((Fin m → ℂ) × (Fin m → ℂ)) from
-    { toFun := fun q => (fun i => (quaternionCoordinates (q i)).1,
-        fun i => (quaternionCoordinates (q i)).2)
-      invFun := fun z i => quaternionCoordinates.symm (z.1 i, z.2 i)
-      left_inv := by intro q; funext i; exact quaternionCoordinates.symm_apply_apply _
-      right_inv := by intro z; ext i <;> simp
-      map_add' := by intro q r; ext i <;> rfl
-      map_smul' := by intro r q; ext i <;> simp [quaternionCoordinates] }).toContinuousLinearEquiv
-
-@[simp] theorem quaternionTupleCoordinates_fixed {m : ℕ} (z : Fin m → ℂ) :
-    quaternionTupleCoordinates (fun i => Quaternion.ofComplex (z i)) = (z, 0) := by
-  apply Prod.ext
-  · funext i; exact congrArg Prod.fst (quaternionCoordinates_ofComplex (z i))
-  · funext i; exact congrArg Prod.snd (quaternionCoordinates_ofComplex (z i))
-
-@[simp] theorem quaternionTupleCoordinates_symm_fixed {m : ℕ} (z : Fin m → ℂ) :
-    quaternionTupleCoordinates.symm (z, 0) = fun i => Quaternion.ofComplex (z i) := by
-  exact quaternionTupleCoordinates.injective (by rw [ContinuousLinearEquiv.apply_symm_apply, quaternionTupleCoordinates_fixed])
-
-/-- Circle normalization chosen compatibly with quaternion normalization. -/
-def normalizeCircle (z : ℂ) : Circle := by
-  classical
-  exact if hz : z = 0 then 1 else ⟨‖z‖⁻¹ • z, by apply mem_sphere_zero_iff_norm.mpr; simp [hz]⟩
-
-theorem normalizeCircle_coe_of_ne_zero {z : ℂ} (hz : z ≠ 0) :
-    (normalizeCircle z : ℂ) = ‖z‖⁻¹ • z := by simp [normalizeCircle, hz]
-
-theorem normalizeQuaternion_ofComplex (z : ℂ) :
-    normalizeQuaternion (Quaternion.ofComplex z) = circleQuaternion (normalizeCircle z) := by
-  by_cases hz : z = 0
-  · subst z
-    simp [normalizeQuaternion, normalizeCircle]
-  · have hq : Quaternion.ofComplex z ≠ 0 := by
-      intro h
-      have hh := congrArg quaternionCoordinates h
-      apply hz
-      have hz0 := congrArg Prod.fst hh
-      change (quaternionCoordinates (Quaternion.ofComplex z)).1 = (quaternionCoordinates (Quaternion.ofComplex 0)).1 at hz0
-      simpa only [quaternionCoordinates_ofComplex] using hz0
-    apply Subtype.ext
-    rw [normalizeQuaternion_coe_of_ne_zero hq, circleQuaternion_coe,
-      normalizeCircle_coe_of_ne_zero hz, norm_quaternion_ofComplex]
-    exact (map_smul Quaternion.ofComplex (‖z‖⁻¹) z).symm
-
-theorem wordMap_circleQuaternion {m : ℕ} (w : Fin m → FreeGroup (Fin m)) (x : Fin m → Circle) :
-    wordMap w (fun j => circleQuaternion (x j)) =
-      fun i => circleQuaternion (∏ j, x j ^ exponentMatrix w i j) := by
-  funext i
-  have he : FreeGroup.lift (fun j => circleQuaternion (x j)) =
-      circleQuaternion.comp (FreeGroup.lift x) := by
-    apply FreeGroup.ext_hom
-    intro j
-    simp
-  change FreeGroup.lift (fun j => circleQuaternion (x j)) (w i) = _
-  rw [he, MonoidHom.comp_apply, freeGroup_lift_commutative]
-  rfl
-
-theorem radialWordMap_fixed {m : ℕ} (w : Fin m → FreeGroup (Fin m)) (z : Fin m → ℂ)
-    (hz : ∀ i, z i ≠ 0) :
-    radialWordMap w (fun i => Quaternion.ofComplex (z i)) =
-      fun i => Quaternion.ofComplex (complexRadialTorus (exponentMatrix w) z i) := by
-  funext i
-  simp only [radialWordMap, normalizeQuaternion_ofComplex, wordMap_circleQuaternion,
-    circleQuaternion_coe, complexRadialTorus, norm_quaternion_ofComplex]
-  rw [map_smul]
-  congr 1
-  apply congrArg Quaternion.ofComplex
-  change Circle.coeHom (∏ j, normalizeCircle (z j) ^ exponentMatrix w i j) = _
-  rw [map_prod]
-  apply Finset.prod_congr rfl
-  intro j _
-  change (normalizeCircle (z j) : ℂ) ^ exponentMatrix w i j = _
-  rw [normalizeCircle_coe_of_ne_zero (hz j)]
-
-theorem quaternionCoordinates_conjComplex (c : ℂ) (q : ℍ) :
-    quaternionCoordinates (Quaternion.ofComplex c * q * star (Quaternion.ofComplex c)) =
-      ((Complex.normSq c : ℂ) * (quaternionCoordinates q).1,
-        c ^ 2 * (quaternionCoordinates q).2) := by
-  apply Prod.ext <;> apply Complex.ext <;>
-    simp [quaternionCoordinates_apply, Complex.normSq_apply, pow_two] <;> ring
-
-@[simp] theorem quaternionCoordinates_conjCircle (c : Circle) (q : ℍ) :
-    quaternionCoordinates (Quaternion.ofComplex c * q * star (Quaternion.ofComplex c)) =
-      ((quaternionCoordinates q).1, (c : ℂ) ^ 2 * (quaternionCoordinates q).2) := by
-  simpa using quaternionCoordinates_conjComplex (c : ℂ) q
-
-/-- Ambient conjugation, compatible with the group operation on unit quaternions. -/
-def quaternionConjugate (a : SU2) (q : ℍ) : ℍ := (a : ℍ) * q * star (a : ℍ)
-
-@[simp] theorem quaternionConjugate_norm (a : SU2) (q : ℍ) :
-    ‖quaternionConjugate a q‖ = ‖q‖ := by
-  have ha := (quaternion_mem_unitary_iff _).mp a.property
-  simp [quaternionConjugate, ha]
-
-@[simp] theorem quaternionConjugate_zero (a : SU2) : quaternionConjugate a 0 = 0 := by
-  simp [quaternionConjugate]
-
-@[simp] theorem quaternionConjugate_one (a : SU2) : quaternionConjugate a 1 = 1 := by
-  simp [quaternionConjugate]
-
-theorem quaternionConjugate_smul (a : SU2) (r : ℝ) (q : ℍ) :
-    quaternionConjugate a (r • q) = r • quaternionConjugate a q := by
-  simp only [quaternionConjugate, mul_smul_comm, smul_mul_assoc]
-
-@[simp] theorem quaternionConjugate_coe (a q : SU2) :
-    quaternionConjugate a (q : ℍ) = ((a * q * a⁻¹ : SU2) : ℍ) := rfl
-
-theorem normalizeQuaternion_conjugate (a : SU2) (q : ℍ) :
-    normalizeQuaternion (quaternionConjugate a q) = a * normalizeQuaternion q * a⁻¹ := by
-  by_cases hq : q = 0
-  · subst q; simp [normalizeQuaternion]
-  · have hn : quaternionConjugate a q ≠ 0 := by
-      intro hz
-      have he := quaternionConjugate_norm a q
-      rw [hz, norm_zero, eq_comm, norm_eq_zero] at he
-      exact hq he
-    apply Subtype.ext
-    rw [normalizeQuaternion_coe_of_ne_zero hn, ← quaternionConjugate_coe,
-      normalizeQuaternion_coe_of_ne_zero hq, quaternionConjugate_smul, quaternionConjugate_norm]
-
-theorem wordMap_conjugation {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (a : SU2) (x : Fin m → SU2) :
-    wordMap w (fun i => a * x i * a⁻¹) = fun i => a * wordMap w x i * a⁻¹ := by
-  have he : FreeGroup.lift (fun i => a * x i * a⁻¹) =
-      (MulAut.conj a).toMonoidHom.comp (FreeGroup.lift x) := by
-    apply FreeGroup.ext_hom
-    intro i
-    simp
-  funext i
-  change FreeGroup.lift (fun i => a * x i * a⁻¹) (w i) = _
-  rw [he]
-  rfl
-
-theorem radialWordMap_conjugation {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (a : SU2) (x : Fin m → ℍ) :
-    radialWordMap w (fun i => quaternionConjugate a (x i)) =
-      fun i => quaternionConjugate a (radialWordMap w x i) := by
-  funext i
-  simp only [radialWordMap, quaternionConjugate_norm, normalizeQuaternion_conjugate,
-    wordMap_conjugation, quaternionConjugate_smul, quaternionConjugate_coe]
-
-theorem smoothRadialWordMap_conjugation {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (a : SU2) (x : Fin m → ℍ) :
-    smoothRadialWordMap w (fun i => quaternionConjugate a (x i)) =
-      fun i => quaternionConjugate a (smoothRadialWordMap w x i) := by
-  funext i
-  simp only [smoothRadialWordMap, radialCutoff, quaternionConjugate_norm,
-    Pi.smul_apply, radialWordMap_conjugation, quaternionConjugate_smul]
-
-end EilenbergGanea
-
-
-namespace EilenbergGanea
-
-theorem complex_root_of_unity_norm {p : ℕ} (hp : p ≠ 0) (z : rootsOfUnity p ℂ) :
-    ‖((z : ℂˣ) : ℂ)‖ = 1 := by
-  have h := congrArg norm ((mem_rootsOfUnity' p (z : ℂˣ)).mp z.property)
-  simpa only [norm_pow, norm_one, pow_eq_one_iff_of_nonneg (norm_nonneg _) hp] using h
-
-theorem rootsOfUnity_isPGroup {p : ℕ} [Fact p.Prime] : IsPGroup p (rootsOfUnity p ℂ) := by
-  let : NeZero p := ⟨(Fact.out : p.Prime).ne_zero⟩
-  rw [isPGroup_iff_card_dvd_pow, Complex.card_rootsOfUnity]
-  exact ⟨1, by simp⟩
-
-theorem rootsOfUnity_exists_nonreal {p : ℕ} (hp : 2 < p) :
-    ∃ z : rootsOfUnity p ℂ, (((z : ℂˣ) : ℂ)).im ≠ 0 := by
-  have hp0 : p ≠ 0 := by omega
-  let ζ := Complex.exp (2 * Real.pi * Complex.I / p)
-  have hζ : IsPrimitiveRoot ζ p := Complex.isPrimitiveRoot_exp p hp0
-  let u : ℂˣ := Units.mk0 ζ (Complex.exp_ne_zero _)
-  let z : rootsOfUnity p ℂ := ⟨u, (mem_rootsOfUnity' p u).mpr hζ.pow_eq_one⟩
-  refine ⟨z, ?_⟩
-  intro him
-  change ζ.im = 0 at him
-  have hn : ‖ζ‖ = 1 := complex_root_of_unity_norm hp0 z
-  have hs : ζ.re ^ 2 = 1 := by
-    have hh := Complex.normSq_eq_norm_sq ζ
-    simp only [Complex.normSq_apply, him, mul_zero, add_zero, hn, one_pow] at hh
-    nlinarith
-  have hpow : ζ ^ 2 = 1 := by
-    apply Complex.ext <;> simp [pow_two, him]
-    nlinarith
-  have hd := hζ.dvd_of_pow_eq_one 2 hpow
-  have ht := Nat.le_of_dvd (by decide : 0 < 2) hd
-  omega
-
-theorem circle_exists_square_root (z : ℂ) (hz : ‖z‖ = 1) :
-    ∃ c : Circle, (c : ℂ) ^ 2 = z := by
-  obtain ⟨c, hc⟩ := IsAlgClosed.exists_pow_nat_eq z (by decide : 0 < 2)
-  have hn : ‖c‖ = 1 := by
-    have hh := congrArg norm hc
-    rw [norm_pow, hz] at hh
-    nlinarith [norm_nonneg c]
-  exact ⟨⟨c, mem_sphere_zero_iff_norm.mpr hn⟩, hc⟩
-
-end EilenbergGanea
-
-namespace EilenbergGanea
-open Set Filter Topology
-open scoped Quaternion ContDiff
-
-def splitWordMap {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (q : (Fin m → ℂ) × (Fin m → ℂ)) : (Fin m → ℂ) × (Fin m → ℂ) :=
-  quaternionTupleCoordinates (smoothRadialWordMap w (quaternionTupleCoordinates.symm q)) - (1, 0)
-
-theorem splitWordMap_contDiff {m : ℕ} (w : Fin m → FreeGroup (Fin m)) :
-    ContDiff ℝ ∞ (splitWordMap w) :=
-  (quaternionTupleCoordinates.contDiff.comp ((smoothRadialWordMap_contDiff w).comp
-    quaternionTupleCoordinates.symm.contDiff)).sub contDiff_const
-
-theorem quaternionTupleCoordinates_conjCircle {m : ℕ} (c : Circle) (q : Fin m → ℍ) :
-    quaternionTupleCoordinates (fun i => quaternionConjugate (circleQuaternion c) (q i)) =
-      ((quaternionTupleCoordinates q).1, (c : ℂ) ^ 2 • (quaternionTupleCoordinates q).2) := by
-  apply Prod.ext
-  · funext i
-    change (quaternionCoordinates (Quaternion.ofComplex c * q i * star (Quaternion.ofComplex c))).1 = _
-    have hh := congrArg Prod.fst (quaternionCoordinates_conjCircle c (q i))
-    exact hh
-  · funext i
-    change (quaternionCoordinates (Quaternion.ofComplex c * q i * star (Quaternion.ofComplex c))).2 = _
-    have hh := congrArg Prod.snd (quaternionCoordinates_conjCircle c (q i))
-    exact hh
-
-theorem splitWordMap_rotation {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (z : ℂ) (hz : ‖z‖ = 1) (q : (Fin m → ℂ) × (Fin m → ℂ)) :
-    splitWordMap w (q.1, z • q.2) = ((splitWordMap w q).1, z • (splitWordMap w q).2) := by
-  obtain ⟨c, hc⟩ := circle_exists_square_root z hz
-  let x := quaternionTupleCoordinates.symm q
-  have hx : quaternionTupleCoordinates x = q := quaternionTupleCoordinates.apply_symm_apply q
-  have heq : (q.1, z • q.2) = quaternionTupleCoordinates
-      (fun i => quaternionConjugate (circleQuaternion c) (x i)) := by
-    rw [quaternionTupleCoordinates_conjCircle, hc, hx]
-  have hh := congrArg (fun v : Fin m → ℍ => quaternionTupleCoordinates v)
-    (smoothRadialWordMap_conjugation w (circleQuaternion c) x)
-  rw [quaternionTupleCoordinates_conjCircle, hc] at hh
-  have heq' : quaternionTupleCoordinates.symm (q.1, z • q.2) =
-      fun i => quaternionConjugate (circleQuaternion c) (x i) := by
-    rw [heq, ContinuousLinearEquiv.symm_apply_apply]
-  unfold splitWordMap
-  rw [heq', hh]
-  change (_, z • _) - (1, 0) = ((quaternionTupleCoordinates (smoothRadialWordMap w x) - (1, 0)).1,
-    z • (quaternionTupleCoordinates (smoothRadialWordMap w x) - (1, 0)).2)
-  simp
-
-theorem splitWordMap_fixed_eventually {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (x : Fin m → ℂ) (hx : ∀ i, ‖x i‖ = 1) :
-    (fun z => (splitWordMap w (z, 0)).1) =ᶠ[𝓝 x]
-      (fun z => complexRadialTorus (exponentMatrix w) z - 1) := by
-  have hne (i : Fin m) : x i ≠ 0 := by intro h; simpa [h] using hx i
-  have hz : ∀ᶠ z in 𝓝 x, ∀ i, z i ≠ 0 := Filter.eventually_all.mpr
-    (fun i => (continuous_apply i).continuousAt.eventually_ne (hne i))
-  have hcont : Continuous (fun z : Fin m → ℂ => fun i => Quaternion.ofComplex (z i)) :=
-    continuous_pi (fun i => Quaternion.ofComplex.toLinearMap.continuous_of_finiteDimensional.comp
-      (continuous_apply i))
-  have he := (smoothRadialWordMap_eventuallyEq w (fun i => Quaternion.ofComplex (x i))
-    (fun i => (norm_quaternion_ofComplex _).trans (hx i))).comp_tendsto hcont.continuousAt
-  filter_upwards [hz, he] with z hz he
-  change smoothRadialWordMap w (fun i => Quaternion.ofComplex (z i)) =
-    radialWordMap w (fun i => Quaternion.ofComplex (z i)) at he
-  simp only [splitWordMap, quaternionTupleCoordinates_symm_fixed]
-  rw [he, radialWordMap_fixed w z hz, quaternionTupleCoordinates_fixed]
-  rfl
-
-theorem splitWordMap_fixed_det {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (x : Fin m → ℂ) (hx : ∀ i, ‖x i‖ = 1)
-    (hr : ∀ i, ∏ j, x j ^ exponentMatrix w i j = 1) :
-    (fderiv ℝ (fun z => (splitWordMap w (z, 0)).1) x).det =
-      ((exponentMatrix w).det : ℝ) := by
-  rw [(splitWordMap_fixed_eventually w x hx).fderiv_eq (𝕜 := ℝ)]
-  have hne (i : Fin m) : x i ≠ 0 := by intro h; simpa [h] using hx i
-  rw [((complexRadialTorus_differentiableAt (exponentMatrix w) x hne).hasFDerivAt.sub_const 1).fderiv]
-  exact complexRadialTorus_det (exponentMatrix w) x hx hr
-
-end EilenbergGanea
-
-namespace EilenbergGanea
-open Set Filter Topology
-open scoped Quaternion
-
-theorem splitWordMap_zero_iff {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (q : (Fin m → ℂ) × (Fin m → ℂ)) :
-    splitWordMap w q = 0 ↔
-      smoothRadialWordMap w (quaternionTupleCoordinates.symm q) = 1 := by
-  unfold splitWordMap
-  rw [sub_eq_zero]
-  have he : quaternionTupleCoordinates (1 : Fin m → ℍ) = (1, 0) := by
-    change quaternionTupleCoordinates (fun i : Fin m => Quaternion.ofComplex (1 : ℂ)) = _
-    exact quaternionTupleCoordinates_fixed 1
-  rw [← he]
-  exact quaternionTupleCoordinates.injective.eq_iff
-
-theorem splitWordMap_fixed_zero_iff {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (z : Fin m → ℂ) : splitWordMap w (z, 0) = 0 ↔
-      (∀ i, ‖z i‖ = 1) ∧ (∀ i, ∏ j, z j ^ exponentMatrix w i j = 1) := by
-  rw [splitWordMap_zero_iff, quaternionTupleCoordinates_symm_fixed,
-    smoothRadialWordMap_eq_one_iff]
-  constructor
-  · intro h
-    have hn : ∀ i, ‖z i‖ = 1 := fun i => (norm_quaternion_ofComplex (z i)).symm.trans
-      (((radialWordMap_eq_one_iff w _).mp h).1 i)
-    have hz (i : Fin m) : z i ≠ 0 := by intro h; simpa [h] using hn i
-    rw [radialWordMap_fixed w z hz] at h
-    refine ⟨hn, fun i => ?_⟩
-    have hh := congrArg (fun v : Fin m → ℍ => (quaternionCoordinates (v i)).1) h
-    change (quaternionCoordinates (Quaternion.ofComplex _)).1 = (1 : ℂ) at hh
-    simpa only [quaternionCoordinates_ofComplex, complexRadialTorus, hn, inv_one, one_smul] using hh
-  · rintro ⟨hn, hr⟩
-    have hz (i : Fin m) : z i ≠ 0 := by intro h; simpa [h] using hn i
-    rw [radialWordMap_fixed w z hz]
-    funext i
-    simp [complexRadialTorus, hn, hr]
-
-@[simp] theorem splitWordMap_fixed_one {m : ℕ} (w : Fin m → FreeGroup (Fin m)) :
-    splitWordMap w (1, 0) = 0 := by
-  rw [splitWordMap_fixed_zero_iff]
-  simp
-
-end EilenbergGanea
-
-namespace EilenbergGanea
-open Set
-section Torus
-variable {n : Type*} [Fintype n] [DecidableEq n]
-variable {A : Type*} [CommGroup A]
-theorem zpow_finite_sum {ι : Type*} (a : A) (s : Finset ι) (e : ι → ℤ) :
-    a ^ (∑ i ∈ s, e i) = ∏ i ∈ s, a ^ e i := by
-  classical
-  induction s using Finset.induction with
-  | empty => simp
-  | @insert i s hi ih => simp [hi, ih, zpow_add]
-
-def torusMap (B : Matrix n n ℤ) (x : n → A) : n → A :=
-  fun i => ∏ j, x j ^ B i j
-
-omit [DecidableEq n] in
-theorem torusMap_comp (B C : Matrix n n ℤ) (x : n → A) :
-    torusMap B (torusMap C x) = torusMap (B * C) x := by
-  funext i
-  simp only [torusMap, Matrix.mul_apply, ← Finset.prod_zpow, zpow_finite_sum, ← zpow_mul]
-  rw [Finset.prod_comm]
-  congr 1
-  funext j
-  congr 1
-  funext k
-  rw [Int.mul_comm]
-
-theorem torusMap_det_one (B : Matrix n n ℤ) (x : n → A) :
-    torusMap (B.det • (1 : Matrix n n ℤ)) x = fun i => x i ^ B.det := by
-  funext i
-  simp [torusMap, Matrix.one_apply]
-
-theorem torusMap_kernel_pow_det (B : Matrix n n ℤ) (x : n → A)
-    (hx : torusMap B x = 1) (i : n) : x i ^ B.det = 1 := by
-  have he := congrArg (fun y => torusMap B.adjugate y i) hx
-  rw [torusMap_comp, Matrix.adjugate_mul, torusMap_det_one] at he
-  simpa [torusMap] using he
-
-theorem circle_finite_pow_roots {d : ℤ} (hd : d ≠ 0) :
-    {z : Circle | z ^ d = 1}.Finite := by
-  have hr := Polynomial.finite_setOfPred_isRoot
-    (Polynomial.X_pow_sub_C_ne_zero (Nat.pos_of_ne_zero (Int.natAbs_ne_zero.mpr hd)) (1 : ℂ))
-  apply Set.Finite.of_injOn (f := fun z : Circle => (z : ℂ)) _
-    Subtype.val_injective.injOn hr
-  intro z hz
-  have hp : z ^ d.natAbs = 1 := pow_natAbs_eq_one.mpr hz
-  have he : (z : ℂ) ^ d.natAbs = 1 := by
-    simpa using congrArg (fun t : Circle => (t : ℂ)) hp
-  simpa [Polynomial.IsRoot.def] using sub_eq_zero.mpr he
-
-theorem torusMap_finite_kernel (B : Matrix n n ℤ) (hB : B.det ≠ 0) :
-    {x : n → Circle | torusMap B x = 1}.Finite := by
-  apply Set.Finite.subset (Set.Finite.pi (fun _ : n => circle_finite_pow_roots hB))
-  intro x hx i _
-  exact torusMap_kernel_pow_det B x hx i
-
-end Torus
-end EilenbergGanea
-
-namespace EilenbergGanea
-open Set Filter Topology
-
-theorem splitWordMap_fixed_finite {m : ℕ} (w : Fin m → FreeGroup (Fin m))
-    (hB : (exponentMatrix w).det ≠ 0) :
-    {z : Fin m → ℂ | splitWordMap w (z, 0) = 0}.Finite := by
-  classical
-  let e : (Fin m → Circle) → (Fin m → ℂ) := fun x i => (x i : ℂ)
-  have hf := (torusMap_finite_kernel (exponentMatrix w) hB).image e
-  apply hf.subset
-  intro z hz
-  obtain ⟨hn, hr⟩ := (splitWordMap_fixed_zero_iff w z).mp hz
-  let x : Fin m → Circle := fun i => ⟨z i, mem_sphere_zero_iff_norm.mpr (hn i)⟩
-  refine ⟨x, ?_, rfl⟩
-  funext i
-  apply Subtype.ext
-  change Circle.coeHom (∏ j, x j ^ exponentMatrix w i j) = (1 : ℂ)
-  rw [map_prod]
-  simp only [map_zpow]
-  exact hr i
-
-end EilenbergGanea
-
-namespace EilenbergGanea
-open Set Filter Topology
-open scoped ContDiff
-
-/-- The signed-root noncancellation for the actual ambient quaternion word
-map, with no regularity assumed of the original word map. -/
-theorem word_exists_regularization_nonzero_count {m : ℕ}
-    (w : Fin m → FreeGroup (Fin m)) (hB : (exponentMatrix w).det ≠ 0)
-    (K : Set ((Fin m → ℂ) × (Fin m → ℂ))) (hK : IsCompact K)
-    (hKi : ∀ z : ℂ, ‖z‖ = 1 → ∀ q ∈ K, (q.1, z • q.2) ∈ K)
-    (h1 : (1, 0) ∈ K)
-    (L : Set ((Fin m → ℂ) × (Fin m → ℂ))) (hL : IsCompact L) (hKL : K ⊆ L)
-    (ε : ℝ) (hε : 0 < ε) :
-    ∃ g : ((Fin m → ℂ) × (Fin m → ℂ)) → ((Fin m → ℂ) × (Fin m → ℂ)),
-      ContDiff ℝ ∞ g ∧ (∀ q ∈ L, ‖g q - splitWordMap w q‖ < ε) ∧
-      ∃ s : Finset ((Fin m → ℂ) × (Fin m → ℂ)),
-        (∀ q, q ∈ s ↔ q ∈ K ∧ g q = 0) ∧
-        (∀ q ∈ s, (fderiv ℝ g q).det ≠ 0) ∧
-        (∑ q ∈ s, (if 0 < (fderiv ℝ g q).det then 1 else -1 : ℝ)) ≠ 0 := by
-  classical
-  let S := (splitWordMap_fixed_finite w hB).toFinset
-  have hS (z) : z ∈ S ↔ splitWordMap w (z, 0) = 0 := (splitWordMap_fixed_finite w hB).mem_toFinset
-  obtain ⟨p, hpbound, hp⟩ := Nat.exists_infinite_primes (S.card + 3)
-  let : Fact p.Prime := ⟨hp⟩
-  let : NeZero p := ⟨hp.ne_zero⟩
-  let : Fintype (rootsOfUnity p ℂ) := Fintype.ofFinite _
-  obtain ⟨a₀, ha₀⟩ := rootsOfUnity_exists_nonreal (p := p) (by omega)
-  let χ : rootsOfUnity p ℂ →* ℂˣ := (rootsOfUnity p ℂ).subtype
-  apply exists_regularization_nonzero_count (rootsOfUnity_isPGroup (p := p)) χ
-    Subtype.val_injective a₀ ha₀ (splitWordMap w) (splitWordMap_contDiff w)
-    (fun a q => splitWordMap_rotation w (χ a : ℂ) (complex_root_of_unity_norm hp.ne_zero a) q)
-    K hK (fun a q hq => hKi (χ a : ℂ) (complex_root_of_unity_norm hp.ne_zero a) q hq)
-    L hL hKL S (fun z _ hz => (hS z).mpr hz) ⟨1, h1, splitWordMap_fixed_one w⟩ (by omega)
-    ((exponentMatrix w).det : ℝ) (by exact_mod_cast hB) _ ε hε
-  intro z hz
-  obtain ⟨hn, hr⟩ := (splitWordMap_fixed_zero_iff w z).mp ((hS z).mp hz)
-  exact splitWordMap_fixed_det w z hn hr
-
-end EilenbergGanea
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.Regularization
+import Mathlib.Analysis.Quaternion
+import Mathlib.Analysis.Complex.Circle
+import Mathlib.Analysis.Complex.Polynomial.Basic
+import Mathlib.Analysis.Calculus.Deriv.ZPow
+import Mathlib.GroupTheory.FreeGroup.Basic
+import Mathlib.Analysis.SpecialFunctions.SmoothTransition
+
+namespace OAI
+
+noncomputable section
+
+open Set Filter Topology
+open scoped Quaternion ContDiff
+
+open scoped Quaternion
+namespace EilenbergGanea
+abbrev SU2 := unitary ℍ
+
+theorem quaternion_mem_unitary_iff (q : ℍ) : q ∈ unitary ℍ ↔ ‖q‖ = 1 := by
+  rw [Unitary.mem_iff, Quaternion.star_mul_self, Quaternion.self_mul_star, and_self]
+  constructor
+  · intro h
+    have he : Quaternion.normSq q = 1 := Quaternion.coe_inj.mp h
+    rw [Quaternion.normSq_eq_norm_mul_self] at he
+    nlinarith [norm_nonneg q]
+  · intro h
+    simp [Quaternion.normSq_eq_norm_mul_self, h]
+
+instance su2_compactSpace : CompactSpace SU2 := by
+  apply isCompact_iff_compactSpace.mp
+  have hs : (unitary ℍ : Set ℍ) = Metric.sphere 0 1 := by
+    ext q
+    simp [quaternion_mem_unitary_iff]
+  rw [hs]
+  exact isCompact_sphere 0 1
+
+def circleQuaternion : Circle →* SU2 where
+  toFun z := ⟨Quaternion.ofComplex z, by
+    rw [Unitary.mem_iff, Quaternion.star_mul_self, Quaternion.self_mul_star, and_self]
+    have hz := Circle.normSq_coe z
+    have he : Quaternion.normSq (Quaternion.ofComplex z) = 1 := by
+      simpa [Quaternion.normSq_def', Complex.normSq_apply, sq] using hz
+    rw [he]; rfl⟩
+  map_one' := Subtype.ext (map_one Quaternion.ofComplex)
+  map_mul' z w := Subtype.ext (map_mul Quaternion.ofComplex (z : ℂ) (w : ℂ))
+
+@[simp] theorem circleQuaternion_coe (z : Circle) :
+    ((circleQuaternion z : SU2) : ℍ) = (z : ℂ) := rfl
+
+theorem circleQuaternion_injective : Function.Injective circleQuaternion := by
+  intro z w h
+  apply Subtype.ext
+  apply Complex.ext
+  · exact congrArg (fun q : ℍ => q.re) (congrArg Subtype.val h)
+  · exact congrArg (fun q : ℍ => q.imI) (congrArg Subtype.val h)
+
+theorem circleQuaternion_continuous : Continuous circleQuaternion := by
+  apply Continuous.subtype_mk
+  exact Quaternion.ofComplex.toLinearMap.continuous_of_finiteDimensional.comp continuous_subtype_val
+
+def wordMap {m : ℕ} (w : Fin m → FreeGroup (Fin m)) (x : Fin m → SU2) : Fin m → SU2 :=
+  fun i => FreeGroup.lift x (w i)
+
+def exponentMatrix {m : ℕ} (w : Fin m → FreeGroup (Fin m)) : Matrix (Fin m) (Fin m) ℤ :=
+  fun i j => Multiplicative.toAdd
+    (FreeGroup.lift (fun k => Multiplicative.ofAdd (if j = k then (1 : ℤ) else 0)) (w i))
+
+open Set Filter Topology
+open scoped ContDiff
+
+/-- Radial normalization is only used on nonzero quaternions. The harmless
+value at zero makes it a total function without changing that domain. -/
+def normalizeQuaternion (q : ℍ) : SU2 := by
+  classical
+  exact if h : q = 0 then 1 else
+    ⟨‖q‖⁻¹ • q, (quaternion_mem_unitary_iff _).mpr (by
+      simp only [norm_smul, Real.norm_eq_abs, abs_inv, abs_norm]
+      exact inv_mul_cancel₀ (norm_ne_zero_iff.mpr h))⟩
+
+theorem normalizeQuaternion_coe_of_ne_zero {q : ℍ} (hq : q ≠ 0) :
+    (normalizeQuaternion q : ℍ) = ‖q‖⁻¹ • q := by simp [normalizeQuaternion, hq]
+
+theorem normalizeQuaternion_coe_contDiffAt {q : ℍ} (hq : q ≠ 0) :
+    ContDiffAt ℝ ∞ (fun z => (normalizeQuaternion z : ℍ)) q := by
+  have h : ContDiffAt ℝ ∞ (fun z : ℍ => ‖z‖⁻¹ • z) q :=
+    ((contDiffAt_norm ℝ hq).inv (norm_ne_zero_iff.mpr hq)).smul contDiffAt_id
+  apply h.congr_of_eventuallyEq
+  filter_upwards [eventually_ne_nhds hq] with z hz
+  exact normalizeQuaternion_coe_of_ne_zero hz
+
+theorem normalizeQuaternion_continuousAt {q : ℍ} (hq : q ≠ 0) :
+    ContinuousAt normalizeQuaternion q := by
+  exact tendsto_subtype_rng.mpr (normalizeQuaternion_coe_contDiffAt hq).continuousAt
+
+@[simp] theorem normalizeQuaternion_unitary (q : SU2) : normalizeQuaternion (q : ℍ) = q := by
+  have hqnorm : ‖(q : ℍ)‖ = 1 := (quaternion_mem_unitary_iff _).mp q.property
+  have hq : (q : ℍ) ≠ 0 := by intro he; simp [he] at hqnorm
+  apply Subtype.ext
+  simp [normalizeQuaternion_coe_of_ne_zero hq, hqnorm]
+
+/-- Smoothness of coefficient-free word evaluation follows directly by
+induction, using quaternion conjugation for unitary inverses. -/
+theorem freeGroup_lift_coe_contDiffAt {D ι : Type*}
+    [NormedAddCommGroup D] [NormedSpace ℝ D] (w : FreeGroup ι)
+    (a : D) (x : D → ι → SU2)
+    (hx : ∀ i, ContDiffAt ℝ ∞ (fun z => (x z i : ℍ)) a) :
+    ContDiffAt ℝ ∞ (fun z => ((FreeGroup.lift (x z) w : SU2) : ℍ)) a := by
+  induction w using FreeGroup.induction_on with
+  | one => simpa using (contDiffAt_const : ContDiffAt ℝ ∞ (fun _ : D => (1 : ℍ)) a)
+  | of i => simpa using hx i
+  | inv_of i hi =>
+      simp only [map_inv]
+      change ContDiffAt ℝ ∞ (fun z => star ((FreeGroup.lift (x z) (FreeGroup.of i) : SU2) : ℍ)) a
+      let L : ℍ →ₗ[ℝ] ℍ :=
+        { toFun := star
+          map_add' := star_add
+          map_smul' := Quaternion.star_smul }
+      exact L.toContinuousLinearMap.contDiff.contDiffAt.comp a hi
+  | mul v w hv hw => simpa only [map_mul, Submonoid.coe_mul] using hv.mul hw
+
+def radialWordMap {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (x : Fin m → ℍ) : Fin m → ℍ :=
+  fun i => ‖x i‖ • ((wordMap w (fun j => normalizeQuaternion (x j)) i : SU2) : ℍ)
+
+@[simp] theorem radialWordMap_norm {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (x : Fin m → ℍ) (i : Fin m) : ‖radialWordMap w x i‖ = ‖x i‖ := by
+  have h : ‖((wordMap w (fun j => normalizeQuaternion (x j)) i : SU2) : ℍ)‖ = 1 :=
+    (quaternion_mem_unitary_iff _).mp (wordMap w (fun j => normalizeQuaternion (x j)) i).property
+  simp [radialWordMap, norm_smul, h]
+
+theorem radialWordMap_contDiffAt {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (x : Fin m → ℍ) (hx : ∀ i, x i ≠ 0) : ContDiffAt ℝ ∞ (radialWordMap w) x := by
+  apply contDiffAt_pi.mpr
+  intro i
+  exact ((contDiffAt_apply ℝ ℍ i x).norm ℝ (hx i)).smul
+    (freeGroup_lift_coe_contDiffAt (w i) x (fun z j => normalizeQuaternion (z j))
+      (fun j => (normalizeQuaternion_coe_contDiffAt (hx j)).comp (f := fun z : Fin m → ℍ => z j) x (contDiffAt_apply ℝ ℍ j x)))
+
+theorem radialWordMap_eq_one_iff {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (x : Fin m → ℍ) : radialWordMap w x = 1 ↔
+      (∀ i, ‖x i‖ = 1) ∧ wordMap w (fun j => normalizeQuaternion (x j)) = 1 := by
+  constructor
+  · intro h
+    have hn (i : Fin m) : ‖x i‖ = 1 := by
+      have he := congrArg (fun y : Fin m → ℍ => ‖y i‖) h
+      simpa only [radialWordMap_norm, Pi.one_apply, norm_one] using he
+    refine ⟨hn, ?_⟩
+    funext i
+    apply Subtype.ext
+    have he := congrFun h i
+    simpa [radialWordMap, hn] using he
+  · rintro ⟨hn, hw⟩
+    funext i
+    simp [radialWordMap, hn, hw]
+
+
+/-- A conjugation-invariant smooth cutoff, zero near coordinate hyperplanes
+and one near the unit quaternion locus. -/
+def radialCutoff {m : ℕ} (x : Fin m → ℍ) : ℝ :=
+  ∏ i, Real.smoothTransition (4 * ‖x i‖ ^ 2 - 1)
+
+def smoothRadialWordMap {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (x : Fin m → ℍ) : Fin m → ℍ := radialCutoff x • radialWordMap w x
+
+theorem radialCutoff_contDiff {m : ℕ} : ContDiff ℝ ∞ (@radialCutoff m) := by
+  apply contDiff_prod
+  intro i _
+  have hn : ContDiff ℝ ∞ (fun x : Fin m → ℍ => ‖x i‖ ^ 2) :=
+    (contDiff_apply ℝ ℍ i).norm_sq ℝ
+  exact Real.smoothTransition.contDiff.comp ((contDiff_const.mul hn).sub contDiff_const)
+
+theorem radialCutoff_nonneg {m : ℕ} (x : Fin m → ℍ) : 0 ≤ radialCutoff x :=
+  Finset.prod_nonneg (fun _ _ => Real.smoothTransition.nonneg _)
+
+theorem radialCutoff_le_one {m : ℕ} (x : Fin m → ℍ) : radialCutoff x ≤ 1 :=
+  Finset.prod_le_one₀ (fun _ _ => Real.smoothTransition.nonneg _)
+    (fun _ _ => Real.smoothTransition.le_one _)
+
+theorem radialCutoff_eq_one {m : ℕ} (x : Fin m → ℍ) (hx : ∀ i, 1 ≤ ‖x i‖) :
+    radialCutoff x = 1 := by
+  apply Finset.prod_eq_one
+  intro i _
+  apply Real.smoothTransition.one_of_one_le
+  nlinarith [hx i]
+
+theorem radialCutoff_eventually_zero {m : ℕ} (x : Fin m → ℍ) (i : Fin m)
+    (hi : x i = 0) : radialCutoff =ᶠ[𝓝 x] (fun _ => 0) := by
+  have h : ∀ᶠ y in 𝓝 x, ‖y i‖ < (1 / 2 : ℝ) :=
+    ((continuous_apply i).norm.continuousAt.eventually_lt_const (by simp [hi]))
+  filter_upwards [h] with y hy
+  apply Finset.prod_eq_zero (Finset.mem_univ i)
+  apply Real.smoothTransition.zero_of_nonpos
+  nlinarith [norm_nonneg (y i)]
+
+theorem radialCutoff_eventually_one {m : ℕ} (x : Fin m → ℍ)
+    (hx : ∀ i, ‖x i‖ = 1) : radialCutoff =ᶠ[𝓝 x] (fun _ => 1) := by
+  have h (i : Fin m) : ∀ᶠ y in 𝓝 x, (3 / 4 : ℝ) < ‖y i‖ :=
+    ((continuous_apply i).norm.continuousAt.eventually_const_lt (by change (3 / 4 : ℝ) < ‖x i‖; rw [hx i]; norm_num))
+  filter_upwards [Filter.eventually_all.mpr h] with y hy
+  apply Finset.prod_eq_one
+  intro i _
+  apply Real.smoothTransition.one_of_one_le
+  nlinarith [hy i]
+
+theorem smoothRadialWordMap_contDiff {m : ℕ} (w : Fin m → FreeGroup (Fin m)) :
+    ContDiff ℝ ∞ (smoothRadialWordMap w) := by
+  rw [contDiff_iff_contDiffAt]
+  intro x
+  by_cases hx : ∀ i, x i ≠ 0
+  · exact radialCutoff_contDiff.contDiffAt.smul (radialWordMap_contDiffAt w x hx)
+  · push Not at hx
+    obtain ⟨i, hi⟩ := hx
+    apply (contDiffAt_const : ContDiffAt ℝ ∞ (fun _ : Fin m → ℍ => (0 : Fin m → ℍ)) x).congr_of_eventuallyEq
+    filter_upwards [radialCutoff_eventually_zero x i hi] with y hy
+    simp [smoothRadialWordMap, hy]
+
+theorem smoothRadialWordMap_eq_one_iff {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (x : Fin m → ℍ) : smoothRadialWordMap w x = 1 ↔ radialWordMap w x = 1 := by
+  constructor
+  · intro h
+    have hn (i : Fin m) : radialCutoff x * ‖x i‖ = 1 := by
+      have he := congrArg (fun y : Fin m → ℍ => ‖y i‖) h
+      simpa only [smoothRadialWordMap, Pi.smul_apply, norm_smul, Real.norm_eq_abs,
+        abs_of_nonneg (radialCutoff_nonneg x), radialWordMap_norm, Pi.one_apply, norm_one] using he
+    have hlarge (i : Fin m) : 1 ≤ ‖x i‖ := by
+      have hle := mul_le_mul_of_nonneg_right (radialCutoff_le_one x) (norm_nonneg (x i))
+      rw [hn i, one_mul] at hle
+      exact hle
+    simpa only [smoothRadialWordMap, radialCutoff_eq_one x hlarge, one_smul] using h
+  · intro h
+    have hn := (radialWordMap_eq_one_iff w x).mp h
+    simp only [smoothRadialWordMap, radialCutoff_eq_one x (fun i => (hn.1 i).ge), one_smul, h]
+
+theorem smoothRadialWordMap_eventuallyEq {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (x : Fin m → ℍ) (hx : ∀ i, ‖x i‖ = 1) :
+    smoothRadialWordMap w =ᶠ[𝓝 x] radialWordMap w := by
+  filter_upwards [radialCutoff_eventually_one x hx] with y hy
+  simp [smoothRadialWordMap, hy]
+
+end EilenbergGanea
+
+namespace EilenbergGanea
+open Set Filter Topology
+variable {n : Type*} [Fintype n] [DecidableEq n]
+
+def splitComplex : (n → ℂ) ≃ₗ[ℝ] ((n → ℝ) × (n → ℝ)) where
+  toFun z := (fun i => (z i).re, fun i => (z i).im)
+  invFun p i := (p.1 i : ℂ) + (p.2 i : ℂ) * Complex.I
+  left_inv z := by funext i; exact Complex.re_add_im (z i)
+  right_inv p := by ext i <;> simp
+  map_add' a b := by ext i <;> simp
+  map_smul' a b := by ext i <;> simp
+
+def radialAngularLinear (B : Matrix n n ℝ) : (n → ℂ) →L[ℝ] (n → ℂ) :=
+  ((splitComplex (n := n)).symm.toLinearMap.comp
+    (((LinearMap.id : (n → ℝ) →ₗ[ℝ] (n → ℝ)).prodMap B.toLin').comp
+      (splitComplex (n := n)).toLinearMap)).toContinuousLinearMap
+
+@[simp] theorem radialAngularLinear_apply (B : Matrix n n ℝ) (z : n → ℂ) (i : n) :
+    radialAngularLinear B z i = ((z i).re : ℂ) + (∑ j, B i j * (z j).im : ℝ) * Complex.I := rfl
+
+theorem radialAngularLinear_det (B : Matrix n n ℝ) :
+    (radialAngularLinear B).toLinearMap.det = B.det := by
+  change LinearMap.det ((splitComplex (n := n)).symm.toLinearMap ∘ₗ
+    ((LinearMap.id : (n → ℝ) →ₗ[ℝ] (n → ℝ)).prodMap B.toLin') ∘ₗ
+    (splitComplex (n := n)).toLinearMap) = _
+  have he := LinearMap.det_conj
+    ((LinearMap.id : (n → ℝ) →ₗ[ℝ] (n → ℝ)).prodMap B.toLin')
+    (splitComplex (n := n)).symm
+  rw [LinearEquiv.symm_symm] at he
+  rw [he, LinearMap.det_prodMap, LinearMap.det_id, one_mul, LinearMap.det_toLin']
+
+def complexPolarChart (x z : n → ℂ) : n → ℂ := fun i => x i * Complex.exp (z i)
+
+def complexRadialTorus (B : Matrix n n ℤ) (z : n → ℂ) : n → ℂ :=
+  fun i => ‖z i‖ • ∏ j, (‖z j‖⁻¹ • z j) ^ B i j
+
+theorem normalize_complex_mul_exp (x z : ℂ) (hx : ‖x‖ = 1) :
+    ‖x * Complex.exp z‖⁻¹ • (x * Complex.exp z) =
+      x * Complex.exp ((z.im : ℂ) * Complex.I) := by
+  rw [norm_mul, hx, one_mul, Complex.norm_exp, Complex.real_smul, Complex.ofReal_inv]
+  have he : Complex.exp z = (Real.exp z.re : ℂ) *
+      Complex.exp ((z.im : ℂ) * Complex.I) := by
+    rw [Complex.ofReal_exp, ← Complex.exp_add, Complex.re_add_im]
+  rw [he]
+  have hn : (Real.exp z.re : ℂ) ≠ 0 := by exact_mod_cast Real.exp_ne_zero z.re
+  field_simp
+
+ theorem complexRadialTorus_chart (B : Matrix n n ℤ) (x : n → ℂ)
+    (hx : ∀ i, ‖x i‖ = 1) (hroot : ∀ i, ∏ j, x j ^ B i j = 1) :
+    complexRadialTorus B ∘ complexPolarChart x =
+      fun z => fun i => Complex.exp (radialAngularLinear (B.map (Int.castRingHom ℝ)) z i) := by
+  funext z i
+  have hn (j : n) := normalize_complex_mul_exp (x j) (z j) (hx j)
+  simp only [Function.comp_apply, complexRadialTorus, complexPolarChart]
+  simp only [hn]
+  simp only [mul_zpow, Finset.prod_mul_distrib,
+    hroot, one_mul, norm_mul, hx, one_mul, Complex.norm_exp, radialAngularLinear_apply,
+    ← Complex.exp_int_mul]
+  rw [Complex.real_smul, Complex.ofReal_exp, ← Complex.exp_sum, ← Complex.exp_add]
+  congr 1
+  simp only [Matrix.map_apply, Int.coe_castRingHom, Complex.ofReal_sum,
+    Complex.ofReal_mul, Complex.ofReal_intCast]
+  rw [Finset.sum_mul]
+  congr 1
+  apply Finset.sum_congr rfl
+  intro j _
+  ring
+
+
+/-- The real derivative of multiplication by a unit complex tuple has determinant one. -/
+def complexDiagonal (x : n → ℂ) : (n → ℂ) →L[ℝ] (n → ℂ) :=
+  ((Matrix.diagonal x).toLin').toContinuousLinearMap.restrictScalars ℝ
+
+@[simp] theorem complexDiagonal_apply (x z : n → ℂ) (i : n) :
+    complexDiagonal x z i = x i * z i := by simp [complexDiagonal, Matrix.toLin'_apply, Matrix.mulVec, Matrix.diagonal_apply, dotProduct]
+
+theorem complexDiagonal_det (x : n → ℂ) (hx : ∀ i, ‖x i‖ = 1) :
+    (complexDiagonal x).toLinearMap.det = 1 := by
+  change (((Matrix.diagonal x).toLin').restrictScalars ℝ).det = _
+  rw [LinearMap.det_restrictScalars, Algebra.norm_complex_apply, LinearMap.det_toLin',
+    Matrix.det_diagonal]
+  rw [Complex.normSq_eq_norm_sq, norm_prod]
+  simp [hx]
+
+theorem complexPolarChart_hasFDerivAt (x : n → ℂ) :
+    HasFDerivAt (complexPolarChart x) (complexDiagonal x) 0 := by
+  apply hasFDerivAt_pi'.mpr
+  intro i
+  have h := ((ContinuousLinearMap.proj i : (n → ℂ) →L[ℝ] ℂ).hasFDerivAt
+    (x := 0)).cexp.const_mul (x i)
+  change HasFDerivAt (fun z : n → ℂ => x i * Complex.exp (z i)) _ (0 : n → ℂ)
+  have he : (ContinuousLinearMap.proj i : (n → ℂ) →L[ℝ] ℂ).comp (complexDiagonal x) =
+      x i • (ContinuousLinearMap.proj i : (n → ℂ) →L[ℝ] ℂ) := by
+    ext z
+    simp
+  rw [he]
+  simpa using h
+
+theorem exp_radialAngularLinear_hasFDerivAt (B : Matrix n n ℝ) :
+    HasFDerivAt (fun z i => Complex.exp (radialAngularLinear B z i))
+      (radialAngularLinear B) 0 := by
+  apply hasFDerivAt_pi'.mpr
+  intro i
+  simpa using (((ContinuousLinearMap.proj i : (n → ℂ) →L[ℝ] ℂ).comp
+    (radialAngularLinear B)).hasFDerivAt (x := 0)).cexp
+
+theorem complexRadialTorus_differentiableAt (B : Matrix n n ℤ) (x : n → ℂ)
+    (hx : ∀ i, x i ≠ 0) : DifferentiableAt ℝ (complexRadialTorus B) x := by
+  apply differentiableAt_pi.mpr
+  intro i
+  have hn (j : n) : DifferentiableAt ℝ (fun z : n → ℂ => ‖z j‖) x :=
+    (differentiableAt_apply j x).norm ℝ (hx j)
+  apply (hn i).smul
+  have hp (j : n) : DifferentiableAt ℝ (fun z : n → ℂ => (‖z j‖⁻¹ • z j) ^ B i j) x := by
+    have hne : ‖x j‖⁻¹ • x j ≠ 0 :=
+      smul_ne_zero (inv_ne_zero (norm_ne_zero_iff.mpr (hx j))) (hx j)
+    have hz : DifferentiableAt ℝ (fun z : ℂ => z ^ B i j) (‖x j‖⁻¹ • x j) :=
+      ((differentiableAt_zpow (𝕜 := ℂ) (m := B i j)).mpr (Or.inl hne)).restrictScalars ℝ
+    exact hz.comp x (((hn j).inv (norm_ne_zero_iff.mpr (hx j))).smul
+      (differentiableAt_apply j x))
+  exact (HasFDerivAt.finsetProd (fun j _ => (hp j).hasFDerivAt)).differentiableAt
+
+/-- At every torus root, the radial extension has exactly determinant det B.
+The extra radial coordinates contribute the identity, not an additional sign. -/
+theorem complexRadialTorus_det (B : Matrix n n ℤ) (x : n → ℂ)
+    (hx : ∀ i, ‖x i‖ = 1) (hroot : ∀ i, ∏ j, x j ^ B i j = 1) :
+    (fderiv ℝ (complexRadialTorus B) x).det = (B.det : ℝ) := by
+  have hne (i : n) : x i ≠ 0 := by intro h; simpa [h] using hx i
+  have hd := (complexRadialTorus_differentiableAt B x hne).hasFDerivAt
+  have hchart : complexPolarChart x 0 = x := by ext i; simp [complexPolarChart]
+  have hc := (show HasFDerivAt (complexRadialTorus B) (fderiv ℝ (complexRadialTorus B) x)
+      (complexPolarChart x 0) by rw [hchart]; exact hd).comp 0 (complexPolarChart_hasFDerivAt x)
+  have he : (fderiv ℝ (complexRadialTorus B) x).comp (complexDiagonal x) =
+      radialAngularLinear (B.map (Int.castRingHom ℝ)) := by
+    apply hc.unique
+    rw [complexRadialTorus_chart B x hx hroot]
+    exact exp_radialAngularLinear_hasFDerivAt _
+  have hdet := congrArg (fun L : (n → ℂ) →L[ℝ] (n → ℂ) => L.det) he
+  change ((fderiv ℝ (complexRadialTorus B) x).toLinearMap.comp
+    (complexDiagonal x).toLinearMap).det = (radialAngularLinear (B.map (Int.castRingHom ℝ))).toLinearMap.det at hdet
+  rw [LinearMap.det_comp, complexDiagonal_det x hx,
+    mul_one, radialAngularLinear_det] at hdet
+  exact hdet.trans ((Int.castRingHom ℝ).map_det B).symm
+
+end EilenbergGanea
+namespace EilenbergGanea
+section CommutativeWords
+variable {n A : Type*} [Fintype n] [DecidableEq n] [CommGroup A]
+def wordExponents {ι : Type*} [DecidableEq ι] (w : FreeGroup ι) (j : ι) : ℤ :=
+  Multiplicative.toAdd
+    (FreeGroup.lift (fun k => Multiplicative.ofAdd (if j = k then (1 : ℤ) else 0)) w)
+
+@[simp] theorem wordExponents_one {ι : Type*} [DecidableEq ι] (j : ι) : wordExponents (1 : FreeGroup ι) j = 0 := by
+  simp [wordExponents]
+
+@[simp] theorem wordExponents_of {ι : Type*} [DecidableEq ι] (j k : ι) :
+    wordExponents (FreeGroup.of k) j = if j = k then 1 else 0 := by
+  simp [wordExponents]
+
+@[simp] theorem wordExponents_inv {ι : Type*} [DecidableEq ι] (w : FreeGroup ι) (j : ι) :
+    wordExponents w⁻¹ j = -wordExponents w j := by
+  simp [wordExponents]
+
+@[simp] theorem wordExponents_mul {ι : Type*} [DecidableEq ι] (w v : FreeGroup ι) (j : ι) :
+    wordExponents (w * v) j = wordExponents w j + wordExponents v j := by
+  simp [wordExponents]
+
+theorem freeGroup_lift_commutative (x : n → A) (w : FreeGroup n) :
+    FreeGroup.lift x w = ∏ j, x j ^ wordExponents w j := by
+  induction w using FreeGroup.induction_on with
+  | one => simp
+  | of i => simp
+  | inv_of i hi => simp [hi, Finset.prod_inv_distrib]
+  | mul a b ha hb => simp [ha, hb, zpow_add, Finset.prod_mul_distrib]
+
+end CommutativeWords
+end EilenbergGanea
+namespace EilenbergGanea
+open Set Filter Topology
+open scoped Quaternion ContDiff
+
+def quaternionCoordinates : ℍ ≃ₗ[ℝ] ℂ × ℂ where
+  toFun q := (⟨q.re, q.imI⟩, ⟨q.imJ, q.imK⟩)
+  invFun z := ⟨z.1.re, z.1.im, z.2.re, z.2.im⟩
+  left_inv q := by cases q; rfl
+  right_inv z := by rcases z with ⟨⟨a,b⟩,⟨c,d⟩⟩; rfl
+  map_add' q r := by ext <;> rfl
+  map_smul' r q := by apply Prod.ext <;> apply Complex.ext <;> simp
+
+@[simp] theorem quaternionCoordinates_apply (q : ℍ) :
+    quaternionCoordinates q = (⟨q.re, q.imI⟩, ⟨q.imJ, q.imK⟩) := rfl
+
+@[simp] theorem quaternionCoordinates_ofComplex (z : ℂ) :
+    quaternionCoordinates (Quaternion.ofComplex z) = (z, 0) := by
+  ext <;> rfl
+
+@[simp] theorem norm_quaternion_ofComplex (z : ℂ) : ‖Quaternion.ofComplex z‖ = ‖z‖ := by
+  have h : ‖Quaternion.ofComplex z‖ ^ 2 = ‖z‖ ^ 2 := by
+    rw [sq, ← Quaternion.normSq_eq_norm_mul_self, ← Complex.normSq_eq_norm_sq]
+    simp [Quaternion.normSq_def', Complex.normSq_apply, sq]
+  nlinarith [norm_nonneg (Quaternion.ofComplex z), norm_nonneg z]
+
+def quaternionTupleCoordinates {m : ℕ} :
+    (Fin m → ℍ) ≃L[ℝ] ((Fin m → ℂ) × (Fin m → ℂ)) :=
+  (show (Fin m → ℍ) ≃ₗ[ℝ] ((Fin m → ℂ) × (Fin m → ℂ)) from
+    { toFun := fun q => (fun i => (quaternionCoordinates (q i)).1,
+        fun i => (quaternionCoordinates (q i)).2)
+      invFun := fun z i => quaternionCoordinates.symm (z.1 i, z.2 i)
+      left_inv := by intro q; funext i; exact quaternionCoordinates.symm_apply_apply _
+      right_inv := by intro z; ext i <;> simp
+      map_add' := by intro q r; ext i <;> rfl
+      map_smul' := by intro r q; ext i <;> simp [quaternionCoordinates] }).toContinuousLinearEquiv
+
+@[simp] theorem quaternionTupleCoordinates_fixed {m : ℕ} (z : Fin m → ℂ) :
+    quaternionTupleCoordinates (fun i => Quaternion.ofComplex (z i)) = (z, 0) := by
+  apply Prod.ext
+  · funext i; exact congrArg Prod.fst (quaternionCoordinates_ofComplex (z i))
+  · funext i; exact congrArg Prod.snd (quaternionCoordinates_ofComplex (z i))
+
+@[simp] theorem quaternionTupleCoordinates_symm_fixed {m : ℕ} (z : Fin m → ℂ) :
+    quaternionTupleCoordinates.symm (z, 0) = fun i => Quaternion.ofComplex (z i) := by
+  exact quaternionTupleCoordinates.injective (by rw [ContinuousLinearEquiv.apply_symm_apply, quaternionTupleCoordinates_fixed])
+
+/-- Circle normalization chosen compatibly with quaternion normalization. -/
+def normalizeCircle (z : ℂ) : Circle := by
+  classical
+  exact if hz : z = 0 then 1 else ⟨‖z‖⁻¹ • z, by apply mem_sphere_zero_iff_norm.mpr; simp [hz]⟩
+
+theorem normalizeCircle_coe_of_ne_zero {z : ℂ} (hz : z ≠ 0) :
+    (normalizeCircle z : ℂ) = ‖z‖⁻¹ • z := by simp [normalizeCircle, hz]
+
+theorem normalizeQuaternion_ofComplex (z : ℂ) :
+    normalizeQuaternion (Quaternion.ofComplex z) = circleQuaternion (normalizeCircle z) := by
+  by_cases hz : z = 0
+  · subst z
+    simp [normalizeQuaternion, normalizeCircle]
+  · have hq : Quaternion.ofComplex z ≠ 0 := by
+      intro h
+      have hh := congrArg quaternionCoordinates h
+      apply hz
+      have hz0 := congrArg Prod.fst hh
+      change (quaternionCoordinates (Quaternion.ofComplex z)).1 = (quaternionCoordinates (Quaternion.ofComplex 0)).1 at hz0
+      simpa only [quaternionCoordinates_ofComplex] using hz0
+    apply Subtype.ext
+    rw [normalizeQuaternion_coe_of_ne_zero hq, circleQuaternion_coe,
+      normalizeCircle_coe_of_ne_zero hz, norm_quaternion_ofComplex]
+    exact (map_smul Quaternion.ofComplex (‖z‖⁻¹) z).symm
+
+theorem wordMap_circleQuaternion {m : ℕ} (w : Fin m → FreeGroup (Fin m)) (x : Fin m → Circle) :
+    wordMap w (fun j => circleQuaternion (x j)) =
+      fun i => circleQuaternion (∏ j, x j ^ exponentMatrix w i j) := by
+  funext i
+  have he : FreeGroup.lift (fun j => circleQuaternion (x j)) =
+      circleQuaternion.comp (FreeGroup.lift x) := by
+    apply FreeGroup.ext_hom
+    intro j
+    simp
+  change FreeGroup.lift (fun j => circleQuaternion (x j)) (w i) = _
+  rw [he, MonoidHom.comp_apply, freeGroup_lift_commutative]
+  rfl
+
+theorem radialWordMap_fixed {m : ℕ} (w : Fin m → FreeGroup (Fin m)) (z : Fin m → ℂ)
+    (hz : ∀ i, z i ≠ 0) :
+    radialWordMap w (fun i => Quaternion.ofComplex (z i)) =
+      fun i => Quaternion.ofComplex (complexRadialTorus (exponentMatrix w) z i) := by
+  funext i
+  simp only [radialWordMap, normalizeQuaternion_ofComplex, wordMap_circleQuaternion,
+    circleQuaternion_coe, complexRadialTorus, norm_quaternion_ofComplex]
+  rw [map_smul]
+  congr 1
+  apply congrArg Quaternion.ofComplex
+  change Circle.coeHom (∏ j, normalizeCircle (z j) ^ exponentMatrix w i j) = _
+  rw [map_prod]
+  apply Finset.prod_congr rfl
+  intro j _
+  change (normalizeCircle (z j) : ℂ) ^ exponentMatrix w i j = _
+  rw [normalizeCircle_coe_of_ne_zero (hz j)]
+
+theorem quaternionCoordinates_conjComplex (c : ℂ) (q : ℍ) :
+    quaternionCoordinates (Quaternion.ofComplex c * q * star (Quaternion.ofComplex c)) =
+      ((Complex.normSq c : ℂ) * (quaternionCoordinates q).1,
+        c ^ 2 * (quaternionCoordinates q).2) := by
+  apply Prod.ext <;> apply Complex.ext <;>
+    simp [quaternionCoordinates_apply, Complex.normSq_apply, pow_two] <;> ring
+
+@[simp] theorem quaternionCoordinates_conjCircle (c : Circle) (q : ℍ) :
+    quaternionCoordinates (Quaternion.ofComplex c * q * star (Quaternion.ofComplex c)) =
+      ((quaternionCoordinates q).1, (c : ℂ) ^ 2 * (quaternionCoordinates q).2) := by
+  simpa using quaternionCoordinates_conjComplex (c : ℂ) q
+
+/-- Ambient conjugation, compatible with the group operation on unit quaternions. -/
+def quaternionConjugate (a : SU2) (q : ℍ) : ℍ := (a : ℍ) * q * star (a : ℍ)
+
+@[simp] theorem quaternionConjugate_norm (a : SU2) (q : ℍ) :
+    ‖quaternionConjugate a q‖ = ‖q‖ := by
+  have ha := (quaternion_mem_unitary_iff _).mp a.property
+  simp [quaternionConjugate, ha]
+
+@[simp] theorem quaternionConjugate_zero (a : SU2) : quaternionConjugate a 0 = 0 := by
+  simp [quaternionConjugate]
+
+@[simp] theorem quaternionConjugate_one (a : SU2) : quaternionConjugate a 1 = 1 := by
+  simp [quaternionConjugate]
+
+theorem quaternionConjugate_smul (a : SU2) (r : ℝ) (q : ℍ) :
+    quaternionConjugate a (r • q) = r • quaternionConjugate a q := by
+  simp only [quaternionConjugate, mul_smul_comm, smul_mul_assoc]
+
+@[simp] theorem quaternionConjugate_coe (a q : SU2) :
+    quaternionConjugate a (q : ℍ) = ((a * q * a⁻¹ : SU2) : ℍ) := rfl
+
+theorem normalizeQuaternion_conjugate (a : SU2) (q : ℍ) :
+    normalizeQuaternion (quaternionConjugate a q) = a * normalizeQuaternion q * a⁻¹ := by
+  by_cases hq : q = 0
+  · subst q; simp [normalizeQuaternion]
+  · have hn : quaternionConjugate a q ≠ 0 := by
+      intro hz
+      have he := quaternionConjugate_norm a q
+      rw [hz, norm_zero, eq_comm, norm_eq_zero] at he
+      exact hq he
+    apply Subtype.ext
+    rw [normalizeQuaternion_coe_of_ne_zero hn, ← quaternionConjugate_coe,
+      normalizeQuaternion_coe_of_ne_zero hq, quaternionConjugate_smul, quaternionConjugate_norm]
+
+theorem wordMap_conjugation {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (a : SU2) (x : Fin m → SU2) :
+    wordMap w (fun i => a * x i * a⁻¹) = fun i => a * wordMap w x i * a⁻¹ := by
+  have he : FreeGroup.lift (fun i => a * x i * a⁻¹) =
+      (MulAut.conj a).toMonoidHom.comp (FreeGroup.lift x) := by
+    apply FreeGroup.ext_hom
+    intro i
+    simp
+  funext i
+  change FreeGroup.lift (fun i => a * x i * a⁻¹) (w i) = _
+  rw [he]
+  rfl
+
+theorem radialWordMap_conjugation {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (a : SU2) (x : Fin m → ℍ) :
+    radialWordMap w (fun i => quaternionConjugate a (x i)) =
+      fun i => quaternionConjugate a (radialWordMap w x i) := by
+  funext i
+  simp only [radialWordMap, quaternionConjugate_norm, normalizeQuaternion_conjugate,
+    wordMap_conjugation, quaternionConjugate_smul, quaternionConjugate_coe]
+
+theorem smoothRadialWordMap_conjugation {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (a : SU2) (x : Fin m → ℍ) :
+    smoothRadialWordMap w (fun i => quaternionConjugate a (x i)) =
+      fun i => quaternionConjugate a (smoothRadialWordMap w x i) := by
+  funext i
+  simp only [smoothRadialWordMap, radialCutoff, quaternionConjugate_norm,
+    Pi.smul_apply, radialWordMap_conjugation, quaternionConjugate_smul]
+
+end EilenbergGanea
+
+
+namespace EilenbergGanea
+
+theorem complex_root_of_unity_norm {p : ℕ} (hp : p ≠ 0) (z : rootsOfUnity p ℂ) :
+    ‖((z : ℂˣ) : ℂ)‖ = 1 := by
+  have h := congrArg norm ((mem_rootsOfUnity' p (z : ℂˣ)).mp z.property)
+  simpa only [norm_pow, norm_one, pow_eq_one_iff_of_nonneg (norm_nonneg _) hp] using h
+
+theorem rootsOfUnity_isPGroup {p : ℕ} [Fact p.Prime] : IsPGroup p (rootsOfUnity p ℂ) := by
+  let : NeZero p := ⟨(Fact.out : p.Prime).ne_zero⟩
+  rw [isPGroup_iff_card_dvd_pow, Complex.card_rootsOfUnity]
+  exact ⟨1, by simp⟩
+
+theorem rootsOfUnity_exists_nonreal {p : ℕ} (hp : 2 < p) :
+    ∃ z : rootsOfUnity p ℂ, (((z : ℂˣ) : ℂ)).im ≠ 0 := by
+  have hp0 : p ≠ 0 := by omega
+  let ζ := Complex.exp (2 * Real.pi * Complex.I / p)
+  have hζ : IsPrimitiveRoot ζ p := Complex.isPrimitiveRoot_exp p hp0
+  let u : ℂˣ := Units.mk0 ζ (Complex.exp_ne_zero _)
+  let z : rootsOfUnity p ℂ := ⟨u, (mem_rootsOfUnity' p u).mpr hζ.pow_eq_one⟩
+  refine ⟨z, ?_⟩
+  intro him
+  change ζ.im = 0 at him
+  have hn : ‖ζ‖ = 1 := complex_root_of_unity_norm hp0 z
+  have hs : ζ.re ^ 2 = 1 := by
+    have hh := Complex.normSq_eq_norm_sq ζ
+    simp only [Complex.normSq_apply, him, mul_zero, add_zero, hn, one_pow] at hh
+    nlinarith
+  have hpow : ζ ^ 2 = 1 := by
+    apply Complex.ext <;> simp [pow_two, him]
+    nlinarith
+  have hd := hζ.dvd_of_pow_eq_one 2 hpow
+  have ht := Nat.le_of_dvd (by decide : 0 < 2) hd
+  omega
+
+theorem circle_exists_square_root (z : ℂ) (hz : ‖z‖ = 1) :
+    ∃ c : Circle, (c : ℂ) ^ 2 = z := by
+  obtain ⟨c, hc⟩ := IsAlgClosed.exists_pow_nat_eq z (by decide : 0 < 2)
+  have hn : ‖c‖ = 1 := by
+    have hh := congrArg norm hc
+    rw [norm_pow, hz] at hh
+    nlinarith [norm_nonneg c]
+  exact ⟨⟨c, mem_sphere_zero_iff_norm.mpr hn⟩, hc⟩
+
+end EilenbergGanea
+
+namespace EilenbergGanea
+open Set Filter Topology
+open scoped Quaternion ContDiff
+
+def splitWordMap {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (q : (Fin m → ℂ) × (Fin m → ℂ)) : (Fin m → ℂ) × (Fin m → ℂ) :=
+  quaternionTupleCoordinates (smoothRadialWordMap w (quaternionTupleCoordinates.symm q)) - (1, 0)
+
+theorem splitWordMap_contDiff {m : ℕ} (w : Fin m → FreeGroup (Fin m)) :
+    ContDiff ℝ ∞ (splitWordMap w) :=
+  (quaternionTupleCoordinates.contDiff.comp ((smoothRadialWordMap_contDiff w).comp
+    quaternionTupleCoordinates.symm.contDiff)).sub contDiff_const
+
+theorem quaternionTupleCoordinates_conjCircle {m : ℕ} (c : Circle) (q : Fin m → ℍ) :
+    quaternionTupleCoordinates (fun i => quaternionConjugate (circleQuaternion c) (q i)) =
+      ((quaternionTupleCoordinates q).1, (c : ℂ) ^ 2 • (quaternionTupleCoordinates q).2) := by
+  apply Prod.ext
+  · funext i
+    change (quaternionCoordinates (Quaternion.ofComplex c * q i * star (Quaternion.ofComplex c))).1 = _
+    have hh := congrArg Prod.fst (quaternionCoordinates_conjCircle c (q i))
+    exact hh
+  · funext i
+    change (quaternionCoordinates (Quaternion.ofComplex c * q i * star (Quaternion.ofComplex c))).2 = _
+    have hh := congrArg Prod.snd (quaternionCoordinates_conjCircle c (q i))
+    exact hh
+
+theorem splitWordMap_rotation {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (z : ℂ) (hz : ‖z‖ = 1) (q : (Fin m → ℂ) × (Fin m → ℂ)) :
+    splitWordMap w (q.1, z • q.2) = ((splitWordMap w q).1, z • (splitWordMap w q).2) := by
+  obtain ⟨c, hc⟩ := circle_exists_square_root z hz
+  let x := quaternionTupleCoordinates.symm q
+  have hx : quaternionTupleCoordinates x = q := quaternionTupleCoordinates.apply_symm_apply q
+  have heq : (q.1, z • q.2) = quaternionTupleCoordinates
+      (fun i => quaternionConjugate (circleQuaternion c) (x i)) := by
+    rw [quaternionTupleCoordinates_conjCircle, hc, hx]
+  have hh := congrArg (fun v : Fin m → ℍ => quaternionTupleCoordinates v)
+    (smoothRadialWordMap_conjugation w (circleQuaternion c) x)
+  rw [quaternionTupleCoordinates_conjCircle, hc] at hh
+  have heq' : quaternionTupleCoordinates.symm (q.1, z • q.2) =
+      fun i => quaternionConjugate (circleQuaternion c) (x i) := by
+    rw [heq, ContinuousLinearEquiv.symm_apply_apply]
+  unfold splitWordMap
+  rw [heq', hh]
+  change (_, z • _) - (1, 0) = ((quaternionTupleCoordinates (smoothRadialWordMap w x) - (1, 0)).1,
+    z • (quaternionTupleCoordinates (smoothRadialWordMap w x) - (1, 0)).2)
+  simp
+
+theorem splitWordMap_fixed_eventually {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (x : Fin m → ℂ) (hx : ∀ i, ‖x i‖ = 1) :
+    (fun z => (splitWordMap w (z, 0)).1) =ᶠ[𝓝 x]
+      (fun z => complexRadialTorus (exponentMatrix w) z - 1) := by
+  have hne (i : Fin m) : x i ≠ 0 := by intro h; simpa [h] using hx i
+  have hz : ∀ᶠ z in 𝓝 x, ∀ i, z i ≠ 0 := Filter.eventually_all.mpr
+    (fun i => (continuous_apply i).continuousAt.eventually_ne (hne i))
+  have hcont : Continuous (fun z : Fin m → ℂ => fun i => Quaternion.ofComplex (z i)) :=
+    continuous_pi (fun i => Quaternion.ofComplex.toLinearMap.continuous_of_finiteDimensional.comp
+      (continuous_apply i))
+  have he := (smoothRadialWordMap_eventuallyEq w (fun i => Quaternion.ofComplex (x i))
+    (fun i => (norm_quaternion_ofComplex _).trans (hx i))).comp_tendsto hcont.continuousAt
+  filter_upwards [hz, he] with z hz he
+  change smoothRadialWordMap w (fun i => Quaternion.ofComplex (z i)) =
+    radialWordMap w (fun i => Quaternion.ofComplex (z i)) at he
+  simp only [splitWordMap, quaternionTupleCoordinates_symm_fixed]
+  rw [he, radialWordMap_fixed w z hz, quaternionTupleCoordinates_fixed]
+  rfl
+
+theorem splitWordMap_fixed_det {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (x : Fin m → ℂ) (hx : ∀ i, ‖x i‖ = 1)
+    (hr : ∀ i, ∏ j, x j ^ exponentMatrix w i j = 1) :
+    (fderiv ℝ (fun z => (splitWordMap w (z, 0)).1) x).det =
+      ((exponentMatrix w).det : ℝ) := by
+  rw [(splitWordMap_fixed_eventually w x hx).fderiv_eq (𝕜 := ℝ)]
+  have hne (i : Fin m) : x i ≠ 0 := by intro h; simpa [h] using hx i
+  rw [((complexRadialTorus_differentiableAt (exponentMatrix w) x hne).hasFDerivAt.sub_const 1).fderiv]
+  exact complexRadialTorus_det (exponentMatrix w) x hx hr
+
+end EilenbergGanea
+
+namespace EilenbergGanea
+open Set Filter Topology
+open scoped Quaternion
+
+theorem splitWordMap_zero_iff {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (q : (Fin m → ℂ) × (Fin m → ℂ)) :
+    splitWordMap w q = 0 ↔
+      smoothRadialWordMap w (quaternionTupleCoordinates.symm q) = 1 := by
+  unfold splitWordMap
+  rw [sub_eq_zero]
+  have he : quaternionTupleCoordinates (1 : Fin m → ℍ) = (1, 0) := by
+    change quaternionTupleCoordinates (fun i : Fin m => Quaternion.ofComplex (1 : ℂ)) = _
+    exact quaternionTupleCoordinates_fixed 1
+  rw [← he]
+  exact quaternionTupleCoordinates.injective.eq_iff
+
+theorem splitWordMap_fixed_zero_iff {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (z : Fin m → ℂ) : splitWordMap w (z, 0) = 0 ↔
+      (∀ i, ‖z i‖ = 1) ∧ (∀ i, ∏ j, z j ^ exponentMatrix w i j = 1) := by
+  rw [splitWordMap_zero_iff, quaternionTupleCoordinates_symm_fixed,
+    smoothRadialWordMap_eq_one_iff]
+  constructor
+  · intro h
+    have hn : ∀ i, ‖z i‖ = 1 := fun i => (norm_quaternion_ofComplex (z i)).symm.trans
+      (((radialWordMap_eq_one_iff w _).mp h).1 i)
+    have hz (i : Fin m) : z i ≠ 0 := by intro h; simpa [h] using hn i
+    rw [radialWordMap_fixed w z hz] at h
+    refine ⟨hn, fun i => ?_⟩
+    have hh := congrArg (fun v : Fin m → ℍ => (quaternionCoordinates (v i)).1) h
+    change (quaternionCoordinates (Quaternion.ofComplex _)).1 = (1 : ℂ) at hh
+    simpa only [quaternionCoordinates_ofComplex, complexRadialTorus, hn, inv_one, one_smul] using hh
+  · rintro ⟨hn, hr⟩
+    have hz (i : Fin m) : z i ≠ 0 := by intro h; simpa [h] using hn i
+    rw [radialWordMap_fixed w z hz]
+    funext i
+    simp [complexRadialTorus, hn, hr]
+
+@[simp] theorem splitWordMap_fixed_one {m : ℕ} (w : Fin m → FreeGroup (Fin m)) :
+    splitWordMap w (1, 0) = 0 := by
+  rw [splitWordMap_fixed_zero_iff]
+  simp
+
+end EilenbergGanea
+
+namespace EilenbergGanea
+open Set
+section Torus
+variable {n : Type*} [Fintype n] [DecidableEq n]
+variable {A : Type*} [CommGroup A]
+theorem zpow_finite_sum {ι : Type*} (a : A) (s : Finset ι) (e : ι → ℤ) :
+    a ^ (∑ i ∈ s, e i) = ∏ i ∈ s, a ^ e i := by
+  classical
+  induction s using Finset.induction with
+  | empty => simp
+  | @insert i s hi ih => simp [hi, ih, zpow_add]
+
+def torusMap (B : Matrix n n ℤ) (x : n → A) : n → A :=
+  fun i => ∏ j, x j ^ B i j
+
+omit [DecidableEq n] in
+theorem torusMap_comp (B C : Matrix n n ℤ) (x : n → A) :
+    torusMap B (torusMap C x) = torusMap (B * C) x := by
+  funext i
+  simp only [torusMap, Matrix.mul_apply, ← Finset.prod_zpow, zpow_finite_sum, ← zpow_mul]
+  rw [Finset.prod_comm]
+  congr 1
+  funext j
+  congr 1
+  funext k
+  rw [Int.mul_comm]
+
+theorem torusMap_det_one (B : Matrix n n ℤ) (x : n → A) :
+    torusMap (B.det • (1 : Matrix n n ℤ)) x = fun i => x i ^ B.det := by
+  funext i
+  simp [torusMap, Matrix.one_apply]
+
+theorem torusMap_kernel_pow_det (B : Matrix n n ℤ) (x : n → A)
+    (hx : torusMap B x = 1) (i : n) : x i ^ B.det = 1 := by
+  have he := congrArg (fun y => torusMap B.adjugate y i) hx
+  rw [torusMap_comp, Matrix.adjugate_mul, torusMap_det_one] at he
+  simpa [torusMap] using he
+
+theorem circle_finite_pow_roots {d : ℤ} (hd : d ≠ 0) :
+    {z : Circle | z ^ d = 1}.Finite := by
+  have hr := Polynomial.finite_setOfPred_isRoot
+    (Polynomial.X_pow_sub_C_ne_zero (Nat.pos_of_ne_zero (Int.natAbs_ne_zero.mpr hd)) (1 : ℂ))
+  apply Set.Finite.of_injOn (f := fun z : Circle => (z : ℂ)) _
+    Subtype.val_injective.injOn hr
+  intro z hz
+  have hp : z ^ d.natAbs = 1 := pow_natAbs_eq_one.mpr hz
+  have he : (z : ℂ) ^ d.natAbs = 1 := by
+    simpa using congrArg (fun t : Circle => (t : ℂ)) hp
+  simpa [Polynomial.IsRoot.def] using sub_eq_zero.mpr he
+
+theorem torusMap_finite_kernel (B : Matrix n n ℤ) (hB : B.det ≠ 0) :
+    {x : n → Circle | torusMap B x = 1}.Finite := by
+  apply Set.Finite.subset (Set.Finite.pi (fun _ : n => circle_finite_pow_roots hB))
+  intro x hx i _
+  exact torusMap_kernel_pow_det B x hx i
+
+end Torus
+end EilenbergGanea
+
+namespace EilenbergGanea
+open Set Filter Topology
+
+theorem splitWordMap_fixed_finite {m : ℕ} (w : Fin m → FreeGroup (Fin m))
+    (hB : (exponentMatrix w).det ≠ 0) :
+    {z : Fin m → ℂ | splitWordMap w (z, 0) = 0}.Finite := by
+  classical
+  let e : (Fin m → Circle) → (Fin m → ℂ) := fun x i => (x i : ℂ)
+  have hf := (torusMap_finite_kernel (exponentMatrix w) hB).image e
+  apply hf.subset
+  intro z hz
+  obtain ⟨hn, hr⟩ := (splitWordMap_fixed_zero_iff w z).mp hz
+  let x : Fin m → Circle := fun i => ⟨z i, mem_sphere_zero_iff_norm.mpr (hn i)⟩
+  refine ⟨x, ?_, rfl⟩
+  funext i
+  apply Subtype.ext
+  change Circle.coeHom (∏ j, x j ^ exponentMatrix w i j) = (1 : ℂ)
+  rw [map_prod]
+  simp only [map_zpow]
+  exact hr i
+
+end EilenbergGanea
+
+namespace EilenbergGanea
+open Set Filter Topology
+open scoped ContDiff
+
+/-- The signed-root noncancellation for the actual ambient quaternion word
+map, with no regularity assumed of the original word map. -/
+theorem word_exists_regularization_nonzero_count {m : ℕ}
+    (w : Fin m → FreeGroup (Fin m)) (hB : (exponentMatrix w).det ≠ 0)
+    (K : Set ((Fin m → ℂ) × (Fin m → ℂ))) (hK : IsCompact K)
+    (hKi : ∀ z : ℂ, ‖z‖ = 1 → ∀ q ∈ K, (q.1, z • q.2) ∈ K)
+    (h1 : (1, 0) ∈ K)
+    (L : Set ((Fin m → ℂ) × (Fin m → ℂ))) (hL : IsCompact L) (hKL : K ⊆ L)
+    (ε : ℝ) (hε : 0 < ε) :
+    ∃ g : ((Fin m → ℂ) × (Fin m → ℂ)) → ((Fin m → ℂ) × (Fin m → ℂ)),
+      ContDiff ℝ ∞ g ∧ (∀ q ∈ L, ‖g q - splitWordMap w q‖ < ε) ∧
+      ∃ s : Finset ((Fin m → ℂ) × (Fin m → ℂ)),
+        (∀ q, q ∈ s ↔ q ∈ K ∧ g q = 0) ∧
+        (∀ q ∈ s, (fderiv ℝ g q).det ≠ 0) ∧
+        (∑ q ∈ s, (if 0 < (fderiv ℝ g q).det then 1 else -1 : ℝ)) ≠ 0 := by
+  classical
+  let S := (splitWordMap_fixed_finite w hB).toFinset
+  have hS (z) : z ∈ S ↔ splitWordMap w (z, 0) = 0 := (splitWordMap_fixed_finite w hB).mem_toFinset
+  obtain ⟨p, hpbound, hp⟩ := Nat.exists_infinite_primes (S.card + 3)
+  let : Fact p.Prime := ⟨hp⟩
+  let : NeZero p := ⟨hp.ne_zero⟩
+  let : Fintype (rootsOfUnity p ℂ) := Fintype.ofFinite _
+  obtain ⟨a₀, ha₀⟩ := rootsOfUnity_exists_nonreal (p := p) (by omega)
+  let χ : rootsOfUnity p ℂ →* ℂˣ := (rootsOfUnity p ℂ).subtype
+  apply exists_regularization_nonzero_count (rootsOfUnity_isPGroup (p := p)) χ
+    Subtype.val_injective a₀ ha₀ (splitWordMap w) (splitWordMap_contDiff w)
+    (fun a q => splitWordMap_rotation w (χ a : ℂ) (complex_root_of_unity_norm hp.ne_zero a) q)
+    K hK (fun a q hq => hKi (χ a : ℂ) (complex_root_of_unity_norm hp.ne_zero a) q hq)
+    L hL hKL S (fun z _ hz => (hS z).mpr hz) ⟨1, h1, splitWordMap_fixed_one w⟩ (by omega)
+    ((exponentMatrix w).det : ℝ) (by exact_mod_cast hB) _ ε hε
+  intro z hz
+  obtain ⟨hn, hr⟩ := (splitWordMap_fixed_zero_iff w z).mp ((hS z).mp hz)
+  exact splitWordMap_fixed_det w z hn hr
+
+end EilenbergGanea
+
+end
+
+end OAI

lean/OAI/Topology/EilenbergGanea/WordEquations.lean

@@ -1,388 +1,388 @@
-import OAI.Topology.EilenbergGanea.IntegralDegree
-import OAI.Topology.EilenbergGanea.UnitaryWords
-
-namespace OAI
-
-noncomputable section
-
-open Classical Set Filter Topology MeasureTheory
-open scoped Quaternion ContDiff
-
-namespace EilenbergGanea
-open Set Filter Topology
-open scoped ContDiff
-
-/-- A concrete sublevel version of localized degree. The only root isolation
-conditions are a uniform radial bound and exclusion of the threshold; no
-regularity of any root or of the sublevel boundary is assumed. -/
-theorem localized_word_root_sublevel {m : ℕ}
-    (w : Fin m → FreeGroup (Fin m)) (hB : (exponentMatrix w).det ≠ 0)
-    (F : ℝ × ((Fin m → ℂ) × (Fin m → ℂ)) → ((Fin m → ℂ) × (Fin m → ℂ)))
-    (hF : ContDiff ℝ 2 F) (hF0 : ∀ q, F (0, q) = splitWordMap w q)
-    (h : ℝ × ((Fin m → ℂ) × (Fin m → ℂ)) → ℝ) (hh : Continuous h)
-    (δ R : ℝ) (hR : ‖((1 : Fin m → ℂ), (0 : Fin m → ℂ))‖ < R)
-    (h1 : h (0, (1, 0)) < δ)
-    (hi : ∀ z : ℂ, ‖z‖ = 1 → ∀ q, h (0, (q.1, z • q.2)) = h (0, q))
-    (hbound : ∀ q, q.1 ∈ Icc (0 : ℝ) 1 → F q = 0 → ‖q.2‖ < R)
-    (hiso : ∀ q, q.1 ∈ Icc (0 : ℝ) 1 → F q = 0 → h q ≠ δ) :
-    ∃ x, F (1, x) = 0 ∧ h (1, x) < δ := by
-  let E := (Fin m → ℂ) × (Fin m → ℂ)
-  let C : Set (ℝ × E) := {q | ‖q.2‖ < R ∧ h q < δ}
-  have hC : IsOpen C := (isOpen_lt continuous_snd.norm continuous_const).inter
-    (isOpen_lt hh continuous_const)
-  have hcl : closure C ⊆ {q | ‖q.2‖ ≤ R ∧ h q ≤ δ} :=
-    closure_minimal (fun _ hq => ⟨hq.1.le, hq.2.le⟩)
-      ((isClosed_le continuous_snd.norm continuous_const).inter (isClosed_le hh continuous_const))
-  have hslice (t : ℝ) : Bornology.IsBounded {x : E | (t, x) ∈ C} := by
-    apply (Metric.isBounded_closedBall (x := (0 : E)) (r := R)).subset
-    intro x hx
-    exact Metric.mem_closedBall.mpr (by simpa only [dist_zero_right] using hx.1.le)
-  have hBcompact : IsCompact {q : ℝ × E | q ∈ frontier C ∧ q.1 ∈ Icc (0 : ℝ) 1} := by
-    apply (isCompact_Icc.prod (isCompact_closedBall (0 : E) R)).of_isClosed_subset
-      (isClosed_frontier.inter (isClosed_Icc.preimage continuous_fst))
-    intro q hq
-    refine ⟨hq.2, ?_⟩
-    exact Metric.mem_closedBall.mpr (by simpa only [dist_zero_right] using (hcl (frontier_subset_closure hq.1)).1)
-  have hCb : Bornology.IsBounded {x : E | ∃ t ∈ Icc (0 : ℝ) 1, (t, x) ∈ C} := by
-    apply (Metric.isBounded_closedBall (x := (0 : E)) (r := R)).subset
-    rintro x ⟨t, _, hx⟩
-    exact Metric.mem_closedBall.mpr (by simpa only [dist_zero_right] using hx.1.le)
-  have hUi : ∀ z : ℂ, ‖z‖ = 1 → ∀ q ∈ closure {x : E | (0, x) ∈ C},
-      (q.1, z • q.2) ∈ closure {x : E | (0, x) ∈ C} := by
-    intro z hz
-    have hmaps : MapsTo (fun q : E => (q.1, z • q.2)) {x | (0, x) ∈ C} {x | (0, x) ∈ C} := by
-      intro q hq
-      refine ⟨?_, ?_⟩
-      · have hn : ‖(q.1, z • q.2)‖ = ‖q‖ := by
-          change max ‖q.1‖ ‖z • q.2‖ = max ‖q.1‖ ‖q.2‖
-          rw [norm_smul, hz, one_mul]
-        simpa only [hn] using hq.1
-      · simpa only [hi z hz q] using hq.2
-    have hr : Continuous (fun q : E => (q.1, z • q.2)) := by fun_prop
-    exact hmaps.closure_of_continuousOn hr.continuousOn
-  have hb : ∀ q ∈ frontier C, q.1 ∈ Icc (0 : ℝ) 1 → F q ≠ 0 := by
-    intro q hq ht hf
-    have hhδ : h q < δ := lt_of_le_of_ne (hcl (frontier_subset_closure hq)).2 (hiso q ht hf)
-    have hqC : q ∈ C := ⟨hbound q ht hf, hhδ⟩
-    exact hq.2 (by simpa only [hC.interior_eq] using hqC)
-  obtain ⟨x, hx, he⟩ := localized_word_root_persists_moving w hB F hF hF0 C hC
-    (hslice 0).isCompact_closure (hslice 1).isCompact_closure hBcompact hCb hUi ⟨hR, h1⟩ hb
-  exact ⟨x, he, hx.2⟩
-end EilenbergGanea
-namespace EilenbergGanea
-open Set Filter Topology
-open scoped Quaternion ContDiff
-
-/-- Word evaluation with a fixed alphabet of coefficient letters and a finite
-alphabet of unknowns. Coefficients may vary smoothly with a real parameter. -/
-def coefficientWordMap {ι : Type*} {m d : ℕ}
-    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (q : ℝ × (Fin m → SU2)) : Fin d → SU2 :=
-  fun i => FreeGroup.lift (Sum.elim (a q.1) q.2) (w i)
-
-def eraseCoefficients {ι : Type*} {m : ℕ} : FreeGroup (ι ⊕ Fin m) →* FreeGroup (Fin m) :=
-  FreeGroup.lift (Sum.elim (fun _ => 1) FreeGroup.of)
-
-theorem coefficientWordMap_initial {ι : Type*} {m d : ℕ}
-    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (ha : ∀ i, a 0 i = 1) (x : Fin m → SU2) :
-    coefficientWordMap w a (0, x) = fun i => FreeGroup.lift x (eraseCoefficients (w i)) := by
-  have he : FreeGroup.lift (Sum.elim (a 0) x) = (FreeGroup.lift x).comp eraseCoefficients := by
-    apply FreeGroup.ext_hom
-    intro i
-    cases i with
-    | inl i => simp [eraseCoefficients, ha]
-    | inr i => simp [eraseCoefficients]
-  funext i
-  exact congrArg (fun f : FreeGroup (ι ⊕ Fin m) →* SU2 => f (w i)) he
-
-/-- The same conjugation-invariant radial cutoff handles arbitrary coefficient
-words, and does not create roots on the coordinate hyperplanes. -/
-theorem radialCutoff_family_smooth {m : ℕ} {V : Type*}
-    [NormedAddCommGroup V] [NormedSpace ℝ V]
-    (f : ℝ × (Fin m → ℍ) → V)
-    (hf : ∀ q, (∀ i, q.2 i ≠ 0) → ContDiffAt ℝ ∞ f q) :
-    ContDiff ℝ ∞ (fun q => radialCutoff q.2 • f q) := by
-  rw [contDiff_iff_contDiffAt]
-  intro q
-  by_cases hq : ∀ i, q.2 i ≠ 0
-  · exact (radialCutoff_contDiff.comp contDiff_snd).contDiffAt.smul (hf q hq)
-  · push Not at hq
-    obtain ⟨i, hi⟩ := hq
-    apply (contDiffAt_const : ContDiffAt ℝ ∞ (fun _ : ℝ × (Fin m → ℍ) => (0 : V)) q).congr_of_eventuallyEq
-    have he : ∀ᶠ p in 𝓝 q, radialCutoff p.2 = 0 :=
-      continuous_snd.continuousAt (radialCutoff_eventually_zero q.2 i hi)
-    filter_upwards [he] with p hp
-    simp only [hp, zero_smul]
-
-theorem coefficientWordMap_normalized_smoothAt {ι : Type*} {m d : ℕ}
-    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (ha : ∀ i, ContDiff ℝ ∞ (fun t => (a t i : ℍ)))
-    (q : ℝ × (Fin m → ℍ)) (hq : ∀ i, q.2 i ≠ 0) :
-    ContDiffAt ℝ ∞ (fun p : ℝ × (Fin m → ℍ) =>
-      fun i => (coefficientWordMap w a (p.1, fun j => normalizeQuaternion (p.2 j)) i : ℍ)) q := by
-  apply contDiffAt_pi.mpr
-  intro i
-  refine freeGroup_lift_coe_contDiffAt (w i) q
-    (fun parameter => Sum.elim (a parameter.1)
-      (fun index => normalizeQuaternion (parameter.2 index))) ?_
-  intro j
-  cases j with
-  | inl j => exact ((ha j).comp contDiff_fst).contDiffAt
-  | inr j =>
-      change ContDiffAt ℝ ∞ (fun p : ℝ × (Fin m → ℍ) => (normalizeQuaternion (p.2 j) : ℍ)) q
-      exact (normalizeQuaternion_coe_contDiffAt (hq j)).comp (f := fun p : ℝ × (Fin m → ℍ) => p.2 j) q
-        ((contDiff_apply ℝ ℍ j).comp contDiff_snd).contDiffAt
-
-def radialCoefficientWordMap {ι : Type*} {m : ℕ}
-    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (q : ℝ × (Fin m → ℍ)) : Fin m → ℍ :=
-  fun i => ‖q.2 i‖ • (coefficientWordMap w a (q.1, fun j => normalizeQuaternion (q.2 j)) i : ℍ)
-
-def smoothRadialCoefficientWordMap {ι : Type*} {m : ℕ}
-    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (q : ℝ × (Fin m → ℍ)) : Fin m → ℍ :=
-  radialCutoff q.2 • radialCoefficientWordMap w a q
-
-theorem smoothRadialCoefficientWordMap_smooth {ι : Type*} {m : ℕ}
-    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (ha : ∀ i, ContDiff ℝ ∞ (fun t => (a t i : ℍ))) :
-    ContDiff ℝ ∞ (smoothRadialCoefficientWordMap w a) := by
-  apply radialCutoff_family_smooth
-  intro q hq
-  apply contDiffAt_pi.mpr
-  intro i
-  exact (((contDiff_apply ℝ ℍ i).comp contDiff_snd).contDiffAt.norm ℝ (hq i)).smul
-    ((contDiffAt_pi.mp (coefficientWordMap_normalized_smoothAt w a ha q hq)) i)
-
-@[simp] theorem radialCoefficientWordMap_norm {ι : Type*} {m : ℕ}
-    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (q : ℝ × (Fin m → ℍ)) (i : Fin m) :
-    ‖radialCoefficientWordMap w a q i‖ = ‖q.2 i‖ := by
-  have h := (quaternion_mem_unitary_iff _).mp
-    (coefficientWordMap w a (q.1, fun j => normalizeQuaternion (q.2 j)) i).property
-  simp [radialCoefficientWordMap, norm_smul, h]
-
-theorem radialCoefficientWordMap_eq_one_iff {ι : Type*} {m : ℕ}
-    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (q : ℝ × (Fin m → ℍ)) : radialCoefficientWordMap w a q = 1 ↔
-      (∀ i, ‖q.2 i‖ = 1) ∧ coefficientWordMap w a
-        (q.1, fun j => normalizeQuaternion (q.2 j)) = 1 := by
-  constructor
-  · intro h
-    have hn (i : Fin m) : ‖q.2 i‖ = 1 := by
-      have he := congrArg (fun y : Fin m → ℍ => ‖y i‖) h
-      simpa only [radialCoefficientWordMap_norm, Pi.one_apply, norm_one] using he
-    refine ⟨hn, ?_⟩
-    funext i
-    apply Subtype.ext
-    have he := congrFun h i
-    simpa [radialCoefficientWordMap, hn] using he
-  · rintro ⟨hn, hw⟩
-    funext i
-    simp [radialCoefficientWordMap, hn, hw]
-
-theorem smoothRadialCoefficientWordMap_eq_one_iff {ι : Type*} {m : ℕ}
-    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (q : ℝ × (Fin m → ℍ)) : smoothRadialCoefficientWordMap w a q = 1 ↔
-      (∀ i, ‖q.2 i‖ = 1) ∧ coefficientWordMap w a
-        (q.1, fun j => normalizeQuaternion (q.2 j)) = 1 := by
-  rw [← radialCoefficientWordMap_eq_one_iff]
-  constructor
-  · intro h
-    have hn (i : Fin m) : radialCutoff q.2 * ‖q.2 i‖ = 1 := by
-      have he := congrArg (fun y : Fin m → ℍ => ‖y i‖) h
-      simpa only [smoothRadialCoefficientWordMap, Pi.smul_apply, norm_smul, Real.norm_eq_abs,
-        abs_of_nonneg (radialCutoff_nonneg q.2), radialCoefficientWordMap_norm, Pi.one_apply, norm_one] using he
-    have hlarge (i : Fin m) : 1 ≤ ‖q.2 i‖ := by
-      have hle := mul_le_mul_of_nonneg_right (radialCutoff_le_one q.2) (norm_nonneg (q.2 i))
-      rw [hn i, one_mul] at hle
-      exact hle
-    simpa only [smoothRadialCoefficientWordMap, radialCutoff_eq_one q.2 hlarge, one_smul] using h
-  · intro h
-    have hn := (radialCoefficientWordMap_eq_one_iff w a q).mp h
-    simp only [smoothRadialCoefficientWordMap, radialCutoff_eq_one q.2 (fun i => (hn.1 i).ge), one_smul, h]
-
-theorem smoothRadialCoefficientWordMap_initial {ι : Type*} {m : ℕ}
-    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (ha : ∀ i, a 0 i = 1) (x : Fin m → ℍ) :
-    smoothRadialCoefficientWordMap w a (0, x) =
-      smoothRadialWordMap (fun i => eraseCoefficients (w i)) x := by
-  unfold smoothRadialCoefficientWordMap smoothRadialWordMap
-  congr 1
-  funext i
-  dsimp only [radialCoefficientWordMap, radialWordMap]
-  rw [coefficientWordMap_initial w a ha]
-  rfl
-end EilenbergGanea
-
-namespace EilenbergGanea
-open Set Filter Topology
-open scoped Quaternion ContDiff
-
-def cutoffCoefficientProducts {ι : Type*} {m d : ℕ}
-    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (q : ℝ × (Fin m → ℍ)) : Fin d → ℍ :=
-  radialCutoff q.2 • fun i =>
-    (coefficientWordMap w a (q.1, fun j => normalizeQuaternion (q.2 j)) i : ℍ)
-
-theorem cutoffCoefficientProducts_smooth {ι : Type*} {m d : ℕ}
-    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (ha : ∀ i, ContDiff ℝ ∞ (fun t => (a t i : ℍ))) :
-    ContDiff ℝ ∞ (cutoffCoefficientProducts w a) :=
-  radialCutoff_family_smooth _ (coefficientWordMap_normalized_smoothAt w a ha)
-
-theorem cutoffCoefficientProducts_unitary {ι : Type*} {m d : ℕ}
-    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (t : ℝ) (x : Fin m → SU2) :
-    cutoffCoefficientProducts w a (t, fun i => (x i : ℍ)) =
-      fun i => (coefficientWordMap w a (t, x) i : ℍ) := by
-  have hn (i : Fin m) : ‖(x i : ℍ)‖ = 1 := (quaternion_mem_unitary_iff _).mp (x i).property
-  simp only [cutoffCoefficientProducts, radialCutoff_eq_one _ (fun i => (hn i).ge), one_smul,
-    normalizeQuaternion_unitary]
-end EilenbergGanea
-
-namespace EilenbergGanea
-open Set Filter Topology
-open scoped Quaternion ContDiff
-
- theorem quaternionConjugate_sub (a : SU2) (q r : ℍ) :
-    quaternionConjugate a (q - r) = quaternionConjugate a q - quaternionConjugate a r := by
-  simp only [quaternionConjugate, mul_sub, sub_mul]
-
- theorem coefficientWordMap_initial_conjugate {ι : Type*} {m d : ℕ}
-    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (ha : ∀ i, a 0 i = 1) (b : SU2) (x : Fin m → SU2) :
-    coefficientWordMap w a (0, fun i => b * x i * b⁻¹) =
-      fun i => b * coefficientWordMap w a (0, x) i * b⁻¹ := by
-  rw [coefficientWordMap_initial w a ha, coefficientWordMap_initial w a ha]
-  have he : FreeGroup.lift (fun i => b * x i * b⁻¹) =
-      (MulAut.conj b).toMonoidHom.comp (FreeGroup.lift x) := by
-    apply FreeGroup.ext_hom
-    intro i
-    simp
-  funext i
-  rw [he]
-  rfl
-
- theorem cutoffCoefficientProducts_initial_conjugate {ι : Type*} {m d : ℕ}
-    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (ha : ∀ i, a 0 i = 1) (b : SU2) (x : Fin m → ℍ) :
-    cutoffCoefficientProducts w a (0, fun i => quaternionConjugate b (x i)) =
-      fun i => quaternionConjugate b (cutoffCoefficientProducts w a (0, x) i) := by
-  funext i
-  simp only [cutoffCoefficientProducts, radialCutoff, quaternionConjugate_norm,
-    normalizeQuaternion_conjugate, coefficientWordMap_initial_conjugate w a ha,
-    Pi.smul_apply, quaternionConjugate_smul, quaternionConjugate_coe]
-
- theorem cutoffCoefficientProducts_initial_size_invariant {ι : Type*} {m d : ℕ}
-    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
-    (ha : ∀ i, a 0 i = 1) (b : SU2) (x : Fin m → ℍ) :
-    ‖cutoffCoefficientProducts w a (0, fun i => quaternionConjugate b (x i)) - 1‖ =
-      ‖cutoffCoefficientProducts w a (0, x) - 1‖ := by
-  rw [cutoffCoefficientProducts_initial_conjugate w a ha]
-  have hn (i : Fin d) : ‖quaternionConjugate b (cutoffCoefficientProducts w a (0, x) i) - 1‖ =
-      ‖cutoffCoefficientProducts w a (0, x) i - 1‖ := by
-    calc
-      _ = ‖quaternionConjugate b (cutoffCoefficientProducts w a (0, x) i - 1)‖ := by rw [quaternionConjugate_sub, quaternionConjugate_one]
-      _ = _ := quaternionConjugate_norm _ _
-  have hnn (i : Fin d) : ‖quaternionConjugate b (cutoffCoefficientProducts w a (0, x) i) - 1‖₊ =
-      ‖cutoffCoefficientProducts w a (0, x) i - 1‖₊ := NNReal.eq (hn i)
-  simp only [Pi.norm_def, Pi.sub_apply, Pi.one_apply, hnn]
-
-/-- Finite-word-system solvability with an actual distinguished-word sublevel
-isolator. Arbitrarily singular initial word roots are allowed. -/
-theorem coefficient_word_sublevel_solvable {ι : Type*} {m d : ℕ}
-    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (ρ : Fin d → FreeGroup (ι ⊕ Fin m))
-    (a : ℝ → ι → SU2) (ha : ∀ i, ContDiff ℝ ∞ (fun t => (a t i : ℍ)))
-    (ha0 : ∀ i, a 0 i = 1)
-    (hB : (exponentMatrix (fun i => eraseCoefficients (w i))).det ≠ 0)
-    (δ : ℝ) (hδ : 0 < δ)
-    (hiso : ∀ t ∈ Icc (0 : ℝ) 1, ∀ x : Fin m → SU2,
-      coefficientWordMap w a (t, x) = 1 →
-      ‖(fun i => (coefficientWordMap ρ a (t, x) i : ℍ)) - 1‖ ≠ δ) :
-    ∃ x : Fin m → SU2, coefficientWordMap w a (1, x) = 1 ∧
-      ‖(fun i => (coefficientWordMap ρ a (1, x) i : ℍ)) - 1‖ < δ := by
-  let E := (Fin m → ℂ) × (Fin m → ℂ)
-  let e : (Fin m → ℍ) ≃L[ℝ] E := quaternionTupleCoordinates
-  let H : ℝ × E → ℝ × (Fin m → ℍ) := fun q => (q.1, e.symm q.2)
-  have hH : ContDiff ℝ ∞ H := contDiff_fst.prodMk (e.symm.contDiff.comp contDiff_snd)
-  let F : ℝ × E → E := fun q => e (smoothRadialCoefficientWordMap w a (H q)) - (1, 0)
-  let h : ℝ × E → ℝ := fun q => ‖cutoffCoefficientProducts ρ a (H q) - 1‖
-  have hF : ContDiff ℝ 2 F := contDiff_infty.mp
-    ((e.contDiff.comp ((smoothRadialCoefficientWordMap_smooth w a ha).comp hH)).sub contDiff_const) 2
-  have hF0 : ∀ q, F (0, q) = splitWordMap (fun i => eraseCoefficients (w i)) q := by
-    intro q
-    dsimp only [F, H]
-    rw [smoothRadialCoefficientWordMap_initial w a ha0]
-    rfl
-  have hh : Continuous h := (((cutoffCoefficientProducts_smooth ρ a ha).comp hH).continuous.sub continuous_const).norm
-  have he1 : e (1 : Fin m → ℍ) = (1, 0) := by
-    change quaternionTupleCoordinates
-      (fun _ : Fin m => Quaternion.ofComplex (1 : ℂ)) = _
-    exact quaternionTupleCoordinates_fixed 1
-  have hroot (q : ℝ × E) (hq : F q = 0) :
-      (∀ i, ‖e.symm q.2 i‖ = 1) ∧
-      coefficientWordMap w a (q.1, fun i => normalizeQuaternion (e.symm q.2 i)) = 1 := by
-    apply (smoothRadialCoefficientWordMap_eq_one_iff w a (H q)).mp
-    apply e.injective
-    rw [he1]
-    exact sub_eq_zero.mp hq
-  have hnorm (q : ℝ × E) (hq : F q = 0) : ‖q.2‖ ≤ ‖e.toContinuousLinearMap‖ := by
-    have hle : ‖e.symm q.2‖ ≤ 1 := (pi_norm_le_iff_of_nonneg (by positivity)).mpr
-      (fun i => (hroot q hq).1 i |>.le)
-    calc
-      ‖q.2‖ = ‖e (e.symm q.2)‖ := by rw [e.apply_symm_apply]
-      _ ≤ ‖e.toContinuousLinearMap‖ * ‖e.symm q.2‖ := e.toContinuousLinearMap.le_opNorm _
-      _ ≤ ‖e.toContinuousLinearMap‖ := mul_le_of_le_one_right (norm_nonneg e.toContinuousLinearMap) hle
-  have hunit (q : ℝ × E) (hq : F q = 0) :
-      (fun i => ((normalizeQuaternion (e.symm q.2 i) : SU2) : ℍ)) = e.symm q.2 := by
-    funext i
-    have hn := (hroot q hq).1 i
-    rw [normalizeQuaternion_coe_of_ne_zero (by intro hz; simp [hz] at hn), hn, inv_one, one_smul]
-  have hsizeroot (q : ℝ × E) (hq : F q = 0) :
-      h q = ‖(fun i => (coefficientWordMap ρ a
-        (q.1, fun j => normalizeQuaternion (e.symm q.2 j)) i : ℍ)) - 1‖ := by
-    dsimp only [h, H]
-    conv_lhs => rw [← hunit q hq, cutoffCoefficientProducts_unitary]
-  have hi : ∀ z : ℂ, ‖z‖ = 1 → ∀ q : E, h (0, (q.1, z • q.2)) = h (0, q) := by
-    intro z hz q
-    obtain ⟨c, hc⟩ := circle_exists_square_root z hz
-    have heq : e.symm (q.1, z • q.2) =
-        fun i => quaternionConjugate (circleQuaternion c) (e.symm q i) := by
-      apply e.injective
-      rw [e.apply_symm_apply, quaternionTupleCoordinates_conjCircle, hc, e.apply_symm_apply]
-    dsimp only [h, H]
-    rw [heq]
-    exact cutoffCoefficientProducts_initial_size_invariant ρ a ha0 _ _
-  have h1 : h (0, (1, 0)) < δ := by
-    have he : e.symm (1, 0) = 1 := by rw [← he1, e.symm_apply_apply]
-    have hp : coefficientWordMap ρ a (0, (1 : Fin m → SU2)) = 1 := by
-      rw [coefficientWordMap_initial ρ a ha0]
-      have hl : FreeGroup.lift (1 : Fin m → SU2) = 1 := by
-        apply FreeGroup.ext_hom
-        intro i
-        simp
-      funext i
-      rw [hl]
-      rfl
-    dsimp only [h, H]
-    rw [he]
-    have hco : (1 : Fin m → ℍ) = fun i => ((1 : Fin m → SU2) i : ℍ) := rfl
-    rw [hco, cutoffCoefficientProducts_unitary, hp]
-    change ‖(1 : Fin d → ℍ) - 1‖ < δ
-    simpa only [sub_self, norm_zero] using hδ
-  let R : ℝ := ‖e.toContinuousLinearMap‖ + ‖((1 : Fin m → ℂ), (0 : Fin m → ℂ))‖ + 1
-  obtain ⟨q, hq, hhq⟩ := localized_word_root_sublevel
-    (fun i => eraseCoefficients (w i)) hB F hF hF0 h hh δ R
-    (by dsimp only [R]; have hn := norm_nonneg e.toContinuousLinearMap; linarith) h1 hi
-    (fun q _ hq => by dsimp only [R]; have hn := hnorm q hq; have hp := norm_nonneg ((1 : Fin m → ℂ), (0 : Fin m → ℂ)); linarith)
-    (fun q ht hq => by rw [hsizeroot q hq]; exact hiso q.1 ht _ (hroot q hq).2)
-  refine ⟨fun i => normalizeQuaternion (e.symm q i), (hroot (1, q) hq).2, ?_⟩
-  simpa only [hsizeroot (1, q) hq] using hhq
-end EilenbergGanea
-
-
-end
-
-end OAI
+import OAI.Topology.EilenbergGanea.IntegralDegree
+import OAI.Topology.EilenbergGanea.UnitaryWords
+
+namespace OAI
+
+noncomputable section
+
+open Classical Set Filter Topology MeasureTheory
+open scoped Quaternion ContDiff
+
+namespace EilenbergGanea
+open Set Filter Topology
+open scoped ContDiff
+
+/-- A concrete sublevel version of localized degree. The only root isolation
+conditions are a uniform radial bound and exclusion of the threshold; no
+regularity of any root or of the sublevel boundary is assumed. -/
+theorem localized_word_root_sublevel {m : ℕ}
+    (w : Fin m → FreeGroup (Fin m)) (hB : (exponentMatrix w).det ≠ 0)
+    (F : ℝ × ((Fin m → ℂ) × (Fin m → ℂ)) → ((Fin m → ℂ) × (Fin m → ℂ)))
+    (hF : ContDiff ℝ 2 F) (hF0 : ∀ q, F (0, q) = splitWordMap w q)
+    (h : ℝ × ((Fin m → ℂ) × (Fin m → ℂ)) → ℝ) (hh : Continuous h)
+    (δ R : ℝ) (hR : ‖((1 : Fin m → ℂ), (0 : Fin m → ℂ))‖ < R)
+    (h1 : h (0, (1, 0)) < δ)
+    (hi : ∀ z : ℂ, ‖z‖ = 1 → ∀ q, h (0, (q.1, z • q.2)) = h (0, q))
+    (hbound : ∀ q, q.1 ∈ Icc (0 : ℝ) 1 → F q = 0 → ‖q.2‖ < R)
+    (hiso : ∀ q, q.1 ∈ Icc (0 : ℝ) 1 → F q = 0 → h q ≠ δ) :
+    ∃ x, F (1, x) = 0 ∧ h (1, x) < δ := by
+  let E := (Fin m → ℂ) × (Fin m → ℂ)
+  let C : Set (ℝ × E) := {q | ‖q.2‖ < R ∧ h q < δ}
+  have hC : IsOpen C := (isOpen_lt continuous_snd.norm continuous_const).inter
+    (isOpen_lt hh continuous_const)
+  have hcl : closure C ⊆ {q | ‖q.2‖ ≤ R ∧ h q ≤ δ} :=
+    closure_minimal (fun _ hq => ⟨hq.1.le, hq.2.le⟩)
+      ((isClosed_le continuous_snd.norm continuous_const).inter (isClosed_le hh continuous_const))
+  have hslice (t : ℝ) : Bornology.IsBounded {x : E | (t, x) ∈ C} := by
+    apply (Metric.isBounded_closedBall (x := (0 : E)) (r := R)).subset
+    intro x hx
+    exact Metric.mem_closedBall.mpr (by simpa only [dist_zero_right] using hx.1.le)
+  have hBcompact : IsCompact {q : ℝ × E | q ∈ frontier C ∧ q.1 ∈ Icc (0 : ℝ) 1} := by
+    apply (isCompact_Icc.prod (isCompact_closedBall (0 : E) R)).of_isClosed_subset
+      (isClosed_frontier.inter (isClosed_Icc.preimage continuous_fst))
+    intro q hq
+    refine ⟨hq.2, ?_⟩
+    exact Metric.mem_closedBall.mpr (by simpa only [dist_zero_right] using (hcl (frontier_subset_closure hq.1)).1)
+  have hCb : Bornology.IsBounded {x : E | ∃ t ∈ Icc (0 : ℝ) 1, (t, x) ∈ C} := by
+    apply (Metric.isBounded_closedBall (x := (0 : E)) (r := R)).subset
+    rintro x ⟨t, _, hx⟩
+    exact Metric.mem_closedBall.mpr (by simpa only [dist_zero_right] using hx.1.le)
+  have hUi : ∀ z : ℂ, ‖z‖ = 1 → ∀ q ∈ closure {x : E | (0, x) ∈ C},
+      (q.1, z • q.2) ∈ closure {x : E | (0, x) ∈ C} := by
+    intro z hz
+    have hmaps : MapsTo (fun q : E => (q.1, z • q.2)) {x | (0, x) ∈ C} {x | (0, x) ∈ C} := by
+      intro q hq
+      refine ⟨?_, ?_⟩
+      · have hn : ‖(q.1, z • q.2)‖ = ‖q‖ := by
+          change max ‖q.1‖ ‖z • q.2‖ = max ‖q.1‖ ‖q.2‖
+          rw [norm_smul, hz, one_mul]
+        simpa only [hn] using hq.1
+      · simpa only [hi z hz q] using hq.2
+    have hr : Continuous (fun q : E => (q.1, z • q.2)) := by fun_prop
+    exact hmaps.closure_of_continuousOn hr.continuousOn
+  have hb : ∀ q ∈ frontier C, q.1 ∈ Icc (0 : ℝ) 1 → F q ≠ 0 := by
+    intro q hq ht hf
+    have hhδ : h q < δ := lt_of_le_of_ne (hcl (frontier_subset_closure hq)).2 (hiso q ht hf)
+    have hqC : q ∈ C := ⟨hbound q ht hf, hhδ⟩
+    exact hq.2 (by simpa only [hC.interior_eq] using hqC)
+  obtain ⟨x, hx, he⟩ := localized_word_root_persists_moving w hB F hF hF0 C hC
+    (hslice 0).isCompact_closure (hslice 1).isCompact_closure hBcompact hCb hUi ⟨hR, h1⟩ hb
+  exact ⟨x, he, hx.2⟩
+end EilenbergGanea
+namespace EilenbergGanea
+open Set Filter Topology
+open scoped Quaternion ContDiff
+
+/-- Word evaluation with a fixed alphabet of coefficient letters and a finite
+alphabet of unknowns. Coefficients may vary smoothly with a real parameter. -/
+def coefficientWordMap {ι : Type*} {m d : ℕ}
+    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (q : ℝ × (Fin m → SU2)) : Fin d → SU2 :=
+  fun i => FreeGroup.lift (Sum.elim (a q.1) q.2) (w i)
+
+def eraseCoefficients {ι : Type*} {m : ℕ} : FreeGroup (ι ⊕ Fin m) →* FreeGroup (Fin m) :=
+  FreeGroup.lift (Sum.elim (fun _ => 1) FreeGroup.of)
+
+theorem coefficientWordMap_initial {ι : Type*} {m d : ℕ}
+    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (ha : ∀ i, a 0 i = 1) (x : Fin m → SU2) :
+    coefficientWordMap w a (0, x) = fun i => FreeGroup.lift x (eraseCoefficients (w i)) := by
+  have he : FreeGroup.lift (Sum.elim (a 0) x) = (FreeGroup.lift x).comp eraseCoefficients := by
+    apply FreeGroup.ext_hom
+    intro i
+    cases i with
+    | inl i => simp [eraseCoefficients, ha]
+    | inr i => simp [eraseCoefficients]
+  funext i
+  exact congrArg (fun f : FreeGroup (ι ⊕ Fin m) →* SU2 => f (w i)) he
+
+/-- The same conjugation-invariant radial cutoff handles arbitrary coefficient
+words, and does not create roots on the coordinate hyperplanes. -/
+theorem radialCutoff_family_smooth {m : ℕ} {V : Type*}
+    [NormedAddCommGroup V] [NormedSpace ℝ V]
+    (f : ℝ × (Fin m → ℍ) → V)
+    (hf : ∀ q, (∀ i, q.2 i ≠ 0) → ContDiffAt ℝ ∞ f q) :
+    ContDiff ℝ ∞ (fun q => radialCutoff q.2 • f q) := by
+  rw [contDiff_iff_contDiffAt]
+  intro q
+  by_cases hq : ∀ i, q.2 i ≠ 0
+  · exact (radialCutoff_contDiff.comp contDiff_snd).contDiffAt.smul (hf q hq)
+  · push Not at hq
+    obtain ⟨i, hi⟩ := hq
+    apply (contDiffAt_const : ContDiffAt ℝ ∞ (fun _ : ℝ × (Fin m → ℍ) => (0 : V)) q).congr_of_eventuallyEq
+    have he : ∀ᶠ p in 𝓝 q, radialCutoff p.2 = 0 :=
+      continuous_snd.continuousAt (radialCutoff_eventually_zero q.2 i hi)
+    filter_upwards [he] with p hp
+    simp only [hp, zero_smul]
+
+theorem coefficientWordMap_normalized_smoothAt {ι : Type*} {m d : ℕ}
+    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (ha : ∀ i, ContDiff ℝ ∞ (fun t => (a t i : ℍ)))
+    (q : ℝ × (Fin m → ℍ)) (hq : ∀ i, q.2 i ≠ 0) :
+    ContDiffAt ℝ ∞ (fun p : ℝ × (Fin m → ℍ) =>
+      fun i => (coefficientWordMap w a (p.1, fun j => normalizeQuaternion (p.2 j)) i : ℍ)) q := by
+  apply contDiffAt_pi.mpr
+  intro i
+  refine freeGroup_lift_coe_contDiffAt (w i) q
+    (fun parameter => Sum.elim (a parameter.1)
+      (fun index => normalizeQuaternion (parameter.2 index))) ?_
+  intro j
+  cases j with
+  | inl j => exact ((ha j).comp contDiff_fst).contDiffAt
+  | inr j =>
+      change ContDiffAt ℝ ∞ (fun p : ℝ × (Fin m → ℍ) => (normalizeQuaternion (p.2 j) : ℍ)) q
+      exact (normalizeQuaternion_coe_contDiffAt (hq j)).comp (f := fun p : ℝ × (Fin m → ℍ) => p.2 j) q
+        ((contDiff_apply ℝ ℍ j).comp contDiff_snd).contDiffAt
+
+def radialCoefficientWordMap {ι : Type*} {m : ℕ}
+    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (q : ℝ × (Fin m → ℍ)) : Fin m → ℍ :=
+  fun i => ‖q.2 i‖ • (coefficientWordMap w a (q.1, fun j => normalizeQuaternion (q.2 j)) i : ℍ)
+
+def smoothRadialCoefficientWordMap {ι : Type*} {m : ℕ}
+    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (q : ℝ × (Fin m → ℍ)) : Fin m → ℍ :=
+  radialCutoff q.2 • radialCoefficientWordMap w a q
+
+theorem smoothRadialCoefficientWordMap_smooth {ι : Type*} {m : ℕ}
+    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (ha : ∀ i, ContDiff ℝ ∞ (fun t => (a t i : ℍ))) :
+    ContDiff ℝ ∞ (smoothRadialCoefficientWordMap w a) := by
+  apply radialCutoff_family_smooth
+  intro q hq
+  apply contDiffAt_pi.mpr
+  intro i
+  exact (((contDiff_apply ℝ ℍ i).comp contDiff_snd).contDiffAt.norm ℝ (hq i)).smul
+    ((contDiffAt_pi.mp (coefficientWordMap_normalized_smoothAt w a ha q hq)) i)
+
+@[simp] theorem radialCoefficientWordMap_norm {ι : Type*} {m : ℕ}
+    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (q : ℝ × (Fin m → ℍ)) (i : Fin m) :
+    ‖radialCoefficientWordMap w a q i‖ = ‖q.2 i‖ := by
+  have h := (quaternion_mem_unitary_iff _).mp
+    (coefficientWordMap w a (q.1, fun j => normalizeQuaternion (q.2 j)) i).property
+  simp [radialCoefficientWordMap, norm_smul, h]
+
+theorem radialCoefficientWordMap_eq_one_iff {ι : Type*} {m : ℕ}
+    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (q : ℝ × (Fin m → ℍ)) : radialCoefficientWordMap w a q = 1 ↔
+      (∀ i, ‖q.2 i‖ = 1) ∧ coefficientWordMap w a
+        (q.1, fun j => normalizeQuaternion (q.2 j)) = 1 := by
+  constructor
+  · intro h
+    have hn (i : Fin m) : ‖q.2 i‖ = 1 := by
+      have he := congrArg (fun y : Fin m → ℍ => ‖y i‖) h
+      simpa only [radialCoefficientWordMap_norm, Pi.one_apply, norm_one] using he
+    refine ⟨hn, ?_⟩
+    funext i
+    apply Subtype.ext
+    have he := congrFun h i
+    simpa [radialCoefficientWordMap, hn] using he
+  · rintro ⟨hn, hw⟩
+    funext i
+    simp [radialCoefficientWordMap, hn, hw]
+
+theorem smoothRadialCoefficientWordMap_eq_one_iff {ι : Type*} {m : ℕ}
+    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (q : ℝ × (Fin m → ℍ)) : smoothRadialCoefficientWordMap w a q = 1 ↔
+      (∀ i, ‖q.2 i‖ = 1) ∧ coefficientWordMap w a
+        (q.1, fun j => normalizeQuaternion (q.2 j)) = 1 := by
+  rw [← radialCoefficientWordMap_eq_one_iff]
+  constructor
+  · intro h
+    have hn (i : Fin m) : radialCutoff q.2 * ‖q.2 i‖ = 1 := by
+      have he := congrArg (fun y : Fin m → ℍ => ‖y i‖) h
+      simpa only [smoothRadialCoefficientWordMap, Pi.smul_apply, norm_smul, Real.norm_eq_abs,
+        abs_of_nonneg (radialCutoff_nonneg q.2), radialCoefficientWordMap_norm, Pi.one_apply, norm_one] using he
+    have hlarge (i : Fin m) : 1 ≤ ‖q.2 i‖ := by
+      have hle := mul_le_mul_of_nonneg_right (radialCutoff_le_one q.2) (norm_nonneg (q.2 i))
+      rw [hn i, one_mul] at hle
+      exact hle
+    simpa only [smoothRadialCoefficientWordMap, radialCutoff_eq_one q.2 hlarge, one_smul] using h
+  · intro h
+    have hn := (radialCoefficientWordMap_eq_one_iff w a q).mp h
+    simp only [smoothRadialCoefficientWordMap, radialCutoff_eq_one q.2 (fun i => (hn.1 i).ge), one_smul, h]
+
+theorem smoothRadialCoefficientWordMap_initial {ι : Type*} {m : ℕ}
+    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (ha : ∀ i, a 0 i = 1) (x : Fin m → ℍ) :
+    smoothRadialCoefficientWordMap w a (0, x) =
+      smoothRadialWordMap (fun i => eraseCoefficients (w i)) x := by
+  unfold smoothRadialCoefficientWordMap smoothRadialWordMap
+  congr 1
+  funext i
+  dsimp only [radialCoefficientWordMap, radialWordMap]
+  rw [coefficientWordMap_initial w a ha]
+  rfl
+end EilenbergGanea
+
+namespace EilenbergGanea
+open Set Filter Topology
+open scoped Quaternion ContDiff
+
+def cutoffCoefficientProducts {ι : Type*} {m d : ℕ}
+    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (q : ℝ × (Fin m → ℍ)) : Fin d → ℍ :=
+  radialCutoff q.2 • fun i =>
+    (coefficientWordMap w a (q.1, fun j => normalizeQuaternion (q.2 j)) i : ℍ)
+
+theorem cutoffCoefficientProducts_smooth {ι : Type*} {m d : ℕ}
+    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (ha : ∀ i, ContDiff ℝ ∞ (fun t => (a t i : ℍ))) :
+    ContDiff ℝ ∞ (cutoffCoefficientProducts w a) :=
+  radialCutoff_family_smooth _ (coefficientWordMap_normalized_smoothAt w a ha)
+
+theorem cutoffCoefficientProducts_unitary {ι : Type*} {m d : ℕ}
+    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (t : ℝ) (x : Fin m → SU2) :
+    cutoffCoefficientProducts w a (t, fun i => (x i : ℍ)) =
+      fun i => (coefficientWordMap w a (t, x) i : ℍ) := by
+  have hn (i : Fin m) : ‖(x i : ℍ)‖ = 1 := (quaternion_mem_unitary_iff _).mp (x i).property
+  simp only [cutoffCoefficientProducts, radialCutoff_eq_one _ (fun i => (hn i).ge), one_smul,
+    normalizeQuaternion_unitary]
+end EilenbergGanea
+
+namespace EilenbergGanea
+open Set Filter Topology
+open scoped Quaternion ContDiff
+
+ theorem quaternionConjugate_sub (a : SU2) (q r : ℍ) :
+    quaternionConjugate a (q - r) = quaternionConjugate a q - quaternionConjugate a r := by
+  simp only [quaternionConjugate, mul_sub, sub_mul]
+
+ theorem coefficientWordMap_initial_conjugate {ι : Type*} {m d : ℕ}
+    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (ha : ∀ i, a 0 i = 1) (b : SU2) (x : Fin m → SU2) :
+    coefficientWordMap w a (0, fun i => b * x i * b⁻¹) =
+      fun i => b * coefficientWordMap w a (0, x) i * b⁻¹ := by
+  rw [coefficientWordMap_initial w a ha, coefficientWordMap_initial w a ha]
+  have he : FreeGroup.lift (fun i => b * x i * b⁻¹) =
+      (MulAut.conj b).toMonoidHom.comp (FreeGroup.lift x) := by
+    apply FreeGroup.ext_hom
+    intro i
+    simp
+  funext i
+  rw [he]
+  rfl
+
+ theorem cutoffCoefficientProducts_initial_conjugate {ι : Type*} {m d : ℕ}
+    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (ha : ∀ i, a 0 i = 1) (b : SU2) (x : Fin m → ℍ) :
+    cutoffCoefficientProducts w a (0, fun i => quaternionConjugate b (x i)) =
+      fun i => quaternionConjugate b (cutoffCoefficientProducts w a (0, x) i) := by
+  funext i
+  simp only [cutoffCoefficientProducts, radialCutoff, quaternionConjugate_norm,
+    normalizeQuaternion_conjugate, coefficientWordMap_initial_conjugate w a ha,
+    Pi.smul_apply, quaternionConjugate_smul, quaternionConjugate_coe]
+
+ theorem cutoffCoefficientProducts_initial_size_invariant {ι : Type*} {m d : ℕ}
+    (w : Fin d → FreeGroup (ι ⊕ Fin m)) (a : ℝ → ι → SU2)
+    (ha : ∀ i, a 0 i = 1) (b : SU2) (x : Fin m → ℍ) :
+    ‖cutoffCoefficientProducts w a (0, fun i => quaternionConjugate b (x i)) - 1‖ =
+      ‖cutoffCoefficientProducts w a (0, x) - 1‖ := by
+  rw [cutoffCoefficientProducts_initial_conjugate w a ha]
+  have hn (i : Fin d) : ‖quaternionConjugate b (cutoffCoefficientProducts w a (0, x) i) - 1‖ =
+      ‖cutoffCoefficientProducts w a (0, x) i - 1‖ := by
+    calc
+      _ = ‖quaternionConjugate b (cutoffCoefficientProducts w a (0, x) i - 1)‖ := by rw [quaternionConjugate_sub, quaternionConjugate_one]
+      _ = _ := quaternionConjugate_norm _ _
+  have hnn (i : Fin d) : ‖quaternionConjugate b (cutoffCoefficientProducts w a (0, x) i) - 1‖₊ =
+      ‖cutoffCoefficientProducts w a (0, x) i - 1‖₊ := NNReal.eq (hn i)
+  simp only [Pi.norm_def, Pi.sub_apply, Pi.one_apply, hnn]
+
+/-- Finite-word-system solvability with an actual distinguished-word sublevel
+isolator. Arbitrarily singular initial word roots are allowed. -/
+theorem coefficient_word_sublevel_solvable {ι : Type*} {m d : ℕ}
+    (w : Fin m → FreeGroup (ι ⊕ Fin m)) (ρ : Fin d → FreeGroup (ι ⊕ Fin m))
+    (a : ℝ → ι → SU2) (ha : ∀ i, ContDiff ℝ ∞ (fun t => (a t i : ℍ)))
+    (ha0 : ∀ i, a 0 i = 1)
+    (hB : (exponentMatrix (fun i => eraseCoefficients (w i))).det ≠ 0)
+    (δ : ℝ) (hδ : 0 < δ)
+    (hiso : ∀ t ∈ Icc (0 : ℝ) 1, ∀ x : Fin m → SU2,
+      coefficientWordMap w a (t, x) = 1 →
+      ‖(fun i => (coefficientWordMap ρ a (t, x) i : ℍ)) - 1‖ ≠ δ) :
+    ∃ x : Fin m → SU2, coefficientWordMap w a (1, x) = 1 ∧
+      ‖(fun i => (coefficientWordMap ρ a (1, x) i : ℍ)) - 1‖ < δ := by
+  let E := (Fin m → ℂ) × (Fin m → ℂ)
+  let e : (Fin m → ℍ) ≃L[ℝ] E := quaternionTupleCoordinates
+  let H : ℝ × E → ℝ × (Fin m → ℍ) := fun q => (q.1, e.symm q.2)
+  have hH : ContDiff ℝ ∞ H := contDiff_fst.prodMk (e.symm.contDiff.comp contDiff_snd)
+  let F : ℝ × E → E := fun q => e (smoothRadialCoefficientWordMap w a (H q)) - (1, 0)
+  let h : ℝ × E → ℝ := fun q => ‖cutoffCoefficientProducts ρ a (H q) - 1‖
+  have hF : ContDiff ℝ 2 F := contDiff_infty.mp
+    ((e.contDiff.comp ((smoothRadialCoefficientWordMap_smooth w a ha).comp hH)).sub contDiff_const) 2
+  have hF0 : ∀ q, F (0, q) = splitWordMap (fun i => eraseCoefficients (w i)) q := by
+    intro q
+    dsimp only [F, H]
+    rw [smoothRadialCoefficientWordMap_initial w a ha0]
+    rfl
+  have hh : Continuous h := (((cutoffCoefficientProducts_smooth ρ a ha).comp hH).continuous.sub continuous_const).norm
+  have he1 : e (1 : Fin m → ℍ) = (1, 0) := by
+    change quaternionTupleCoordinates
+      (fun _ : Fin m => Quaternion.ofComplex (1 : ℂ)) = _
+    exact quaternionTupleCoordinates_fixed 1
+  have hroot (q : ℝ × E) (hq : F q = 0) :
+      (∀ i, ‖e.symm q.2 i‖ = 1) ∧
+      coefficientWordMap w a (q.1, fun i => normalizeQuaternion (e.symm q.2 i)) = 1 := by
+    apply (smoothRadialCoefficientWordMap_eq_one_iff w a (H q)).mp
+    apply e.injective
+    rw [he1]
+    exact sub_eq_zero.mp hq
+  have hnorm (q : ℝ × E) (hq : F q = 0) : ‖q.2‖ ≤ ‖e.toContinuousLinearMap‖ := by
+    have hle : ‖e.symm q.2‖ ≤ 1 := (pi_norm_le_iff_of_nonneg (by positivity)).mpr
+      (fun i => (hroot q hq).1 i |>.le)
+    calc
+      ‖q.2‖ = ‖e (e.symm q.2)‖ := by rw [e.apply_symm_apply]
+      _ ≤ ‖e.toContinuousLinearMap‖ * ‖e.symm q.2‖ := e.toContinuousLinearMap.le_opNorm _
+      _ ≤ ‖e.toContinuousLinearMap‖ := mul_le_of_le_one_right (norm_nonneg e.toContinuousLinearMap) hle
+  have hunit (q : ℝ × E) (hq : F q = 0) :
+      (fun i => ((normalizeQuaternion (e.symm q.2 i) : SU2) : ℍ)) = e.symm q.2 := by
+    funext i
+    have hn := (hroot q hq).1 i
+    rw [normalizeQuaternion_coe_of_ne_zero (by intro hz; simp [hz] at hn), hn, inv_one, one_smul]
+  have hsizeroot (q : ℝ × E) (hq : F q = 0) :
+      h q = ‖(fun i => (coefficientWordMap ρ a
+        (q.1, fun j => normalizeQuaternion (e.symm q.2 j)) i : ℍ)) - 1‖ := by
+    dsimp only [h, H]
+    conv_lhs => rw [← hunit q hq, cutoffCoefficientProducts_unitary]
+  have hi : ∀ z : ℂ, ‖z‖ = 1 → ∀ q : E, h (0, (q.1, z • q.2)) = h (0, q) := by
+    intro z hz q
+    obtain ⟨c, hc⟩ := circle_exists_square_root z hz
+    have heq : e.symm (q.1, z • q.2) =
+        fun i => quaternionConjugate (circleQuaternion c) (e.symm q i) := by
+      apply e.injective
+      rw [e.apply_symm_apply, quaternionTupleCoordinates_conjCircle, hc, e.apply_symm_apply]
+    dsimp only [h, H]
+    rw [heq]
+    exact cutoffCoefficientProducts_initial_size_invariant ρ a ha0 _ _
+  have h1 : h (0, (1, 0)) < δ := by
+    have he : e.symm (1, 0) = 1 := by rw [← he1, e.symm_apply_apply]
+    have hp : coefficientWordMap ρ a (0, (1 : Fin m → SU2)) = 1 := by
+      rw [coefficientWordMap_initial ρ a ha0]
+      have hl : FreeGroup.lift (1 : Fin m → SU2) = 1 := by
+        apply FreeGroup.ext_hom
+        intro i
+        simp
+      funext i
+      rw [hl]
+      rfl
+    dsimp only [h, H]
+    rw [he]
+    have hco : (1 : Fin m → ℍ) = fun i => ((1 : Fin m → SU2) i : ℍ) := rfl
+    rw [hco, cutoffCoefficientProducts_unitary, hp]
+    change ‖(1 : Fin d → ℍ) - 1‖ < δ
+    simpa only [sub_self, norm_zero] using hδ
+  let R : ℝ := ‖e.toContinuousLinearMap‖ + ‖((1 : Fin m → ℂ), (0 : Fin m → ℂ))‖ + 1
+  obtain ⟨q, hq, hhq⟩ := localized_word_root_sublevel
+    (fun i => eraseCoefficients (w i)) hB F hF hF0 h hh δ R
+    (by dsimp only [R]; have hn := norm_nonneg e.toContinuousLinearMap; linarith) h1 hi
+    (fun q _ hq => by dsimp only [R]; have hn := hnorm q hq; have hp := norm_nonneg ((1 : Fin m → ℂ), (0 : Fin m → ℂ)); linarith)
+    (fun q ht hq => by rw [hsizeroot q hq]; exact hiso q.1 ht _ (hroot q hq).2)
+  refine ⟨fun i => normalizeQuaternion (e.symm q i), (hroot (1, q) hq).2, ?_⟩
+  simpa only [hsizeroot (1, q) hq] using hhq
+end EilenbergGanea
+
+
+end
+
+end OAI

lean/OAI/Topology/HahnSeries/Pivots.lean

@@ -1,852 +1,852 @@
-import OAI.LinearAlgebra.RankMatching.Basic
-
-namespace OAI
-
-noncomputable section
-open scoped BigOperators nonZeroDivisors
-open LinearMap Submodule
-open CategoryTheory CategoryTheory.Limits HomologicalComplex
-
-namespace HahnWilson.PivotTopology
-open HahnSeries
-open scoped BigOperators Topology
-variable {k : Type*} [Field k]
-
-def poleLattice (J : ℤ) : Submodule k (LaurentSeries k) where
-  carrier := {f | ∀ i : ℤ, J < i → f.coeff (-i) = 0}
-  zero_mem' := by simp
-  add_mem' hf hg i hi := by simp [hf i hi, hg i hi]
-  smul_mem' c f hf i hi := by simp [hf i hi]
-
-def monomial (j : ℤ) : LaurentSeries k := HahnSeries.single (-j) 1
-
-def PreservesPoles (a : LaurentSeries k →ₗ[k] LaurentSeries k) : Prop :=
-  ∀ J, ∀ f ∈ poleLattice (k := k) J, a f ∈ poleLattice (k := k) J
-
-def matrixOf (a : LaurentSeries k →ₗ[k] LaurentSeries k) : Matrix ℤ ℤ k :=
-  fun l j => (a (monomial j)).coeff (-l)
-
-def HasLeadingTerm (f : LaurentSeries k) (j : ℤ) : Prop :=
-  f ∈ poleLattice (k := k) j ∧ f.coeff (-j) = 1
-
-def HasLeadingPosition (f : LaurentSeries k) (j : ℤ) : Prop :=
-  f ∈ poleLattice (k := k) j ∧ f.coeff (-j) ≠ 0
-
-def crossing (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) : Set (ℤ × ℤ) :=
-  {p | RankMatching.Matched (matrixOf a) p.1 p.2 ∧ p.1 ≤ J ∧ J < p.2}
-
-def crossingQuotient (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :=
-  ↥(LinearMap.range a ⊓ poleLattice (k := k) J) ⧸
-    Submodule.comap (LinearMap.range a ⊓ poleLattice (k := k) J).subtype
-      ((poleLattice (k := k) J).map a)
-
-instance crossingQuotient_addCommGroup (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
-    AddCommGroup (crossingQuotient a J) := by
-  unfold crossingQuotient
-  infer_instance
-
-instance crossingQuotient_module (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
-    Module k (crossingQuotient a J) := by
-  unfold crossingQuotient
-  infer_instance
-
-@[simp] lemma mem_poleLattice (J : ℤ) (f : LaurentSeries k) :
-    f ∈ poleLattice J ↔ ∀ i : ℤ, J < i → f.coeff (-i) = 0 := Iff.rfl
-
-lemma poleLattice_mono {I J : ℤ} (h : I ≤ J) :
-    poleLattice (k := k) I ≤ poleLattice J := by
-  intro f hf i hi
-  exact hf i (lt_of_le_of_lt h hi)
-
-@[simp] lemma coeff_monomial (j i : ℤ) :
-    (monomial (k := k) j).coeff (-i) = if i = j then 1 else 0 := by
-  simp [monomial, HahnSeries.coeff_single]
-
-lemma monomial_mem {j J : ℤ} (h : j ≤ J) : monomial (k := k) j ∈ poleLattice J := by
-  intro i hi
-  simp [show i ≠ j by omega]
-
-lemma exists_poleLattice (f : LaurentSeries k) : ∃ J, f ∈ poleLattice J := by
-  refine ⟨-f.order, ?_⟩
-  intro i hi
-  exact HahnSeries.coeff_eq_zero_of_lt_order (by omega)
-
-lemma matrixOf_triangular (a : LaurentSeries k →ₗ[k] LaurentSeries k)
-    (ha : PreservesPoles a) : RankMatching.Triangular (matrixOf a) := by
-  intro l j h
-  exact ha j (monomial j) (monomial_mem le_rfl) l h
-
-def truncate (f : LaurentSeries k) (L J : ℤ) : LaurentSeries k :=
-  ∑ c ∈ Finset.Icc L J, f.coeff (-c) • monomial c
-
-lemma coeff_truncate (f : LaurentSeries k) (L J i : ℤ) :
-    (truncate f L J).coeff (-i) = if L ≤ i ∧ i ≤ J then f.coeff (-i) else 0 := by
-  classical
-  simp [truncate, HahnSeries.coeff_sum, HahnSeries.coeff_smul, smul_eq_mul,
-    mul_ite]
-
-lemma truncate_mem (f : LaurentSeries k) (L J : ℤ) : truncate f L J ∈ poleLattice J := by
-  intro i hi
-  rw [coeff_truncate, ite_eq_right (by omega)]
-
-lemma sub_truncate_mem {f : LaurentSeries k} {L J : ℤ} (hf : f ∈ poleLattice J) :
-    f - truncate f L J ∈ poleLattice (L - 1) := by
-  intro i hi
-  rw [HahnSeries.coeff_sub, coeff_truncate]
-  by_cases hij : i ≤ J
-  · rw [ite_eq_left (by omega), sub_self]
-  · rw [ite_eq_right (by omega), sub_zero, hf i (by omega)]
-
-def poleProjection (L : ℤ) : LaurentSeries k →ₗ[k] (ℤ → k) where
-  toFun f i := if L ≤ i then f.coeff (-i) else 0
-  map_add' f g := by ext i; by_cases hi : L ≤ i <;> simp [hi]
-  map_smul' c f := by ext i; by_cases hi : L ≤ i <;> simp [hi]
-
-lemma poleProjection_eq_zero {L : ℤ} {f : LaurentSeries k} :
-    poleProjection L f = 0 ↔ f ∈ poleLattice (L - 1) := by
-  constructor
-  · intro h i hi
-    have := congrFun h i
-    simpa [poleProjection, show L ≤ i by omega] using this
-  · intro h
-    ext i
-    by_cases hi : L ≤ i
-    · simp [poleProjection, hi, h i (by omega)]
-    · simp [poleProjection, hi]
-
-@[simp] lemma poleProjection_monomial_image
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (L j : ℤ) :
-    poleProjection L (a (monomial j)) = RankMatching.column (matrixOf a) L j := rfl
-
-lemma poleProjection_truncate_image (a : LaurentSeries k →ₗ[k] LaurentSeries k)
-    (ha : PreservesPoles a) {f : LaurentSeries k} {L J : ℤ}
-    (hf : f ∈ poleLattice J) :
-    poleProjection L (a (truncate f L J)) = poleProjection L (a f) := by
-  have h := (poleProjection_eq_zero).mpr (ha (L - 1) _ (sub_truncate_mem hf))
-  rw [map_sub, map_sub, sub_eq_zero] at h
-  exact h.symm
-
-lemma columnSpace_eq_projectedImage (a : LaurentSeries k →ₗ[k] LaurentSeries k)
-    (ha : PreservesPoles a) (L J : ℤ) :
-    RankMatching.columnSpace (matrixOf a) L J =
-      ((poleLattice J).map a).map (poleProjection L) := by
-  classical
-  apply le_antisymm
-  · apply Submodule.span_le.mpr
-    rintro _ ⟨c, hc, rfl⟩
-    exact ⟨a (monomial c), ⟨monomial c, monomial_mem hc, rfl⟩, rfl⟩
-  · rintro y ⟨z, ⟨f,hf,rfl⟩,rfl⟩
-    rw [← poleProjection_truncate_image a ha hf]
-    simp only [truncate, map_sum, map_smul]
-    exact Submodule.sum_mem _ fun c hc =>
-      Submodule.smul_mem _ _ (Submodule.subset_span
-        ⟨c, (Finset.mem_Icc.mp hc).2, rfl⟩)
-
-lemma increment_zero_iff_finite_solution
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (L j : ℤ) :
-    RankMatching.increment (matrixOf a) L j = 0 ↔
-      ∃ f, HasLeadingTerm f j ∧ a f ∈ poleLattice (L - 1) := by
-  rw [RankMatching.increment_eq_zero_iff, columnSpace_eq_projectedImage a ha]
-  constructor
-  · rintro ⟨z, ⟨g,hg,rfl⟩,heq⟩
-    refine ⟨monomial j - g, ⟨?_, ?_⟩, ?_⟩
-    · intro i hi
-      simp only [HahnSeries.coeff_sub, coeff_monomial,
-        ite_eq_right (show i ≠ j by omega), hg i (by omega), sub_self]
-    · simp [HahnSeries.coeff_sub, hg j (by omega)]
-    · apply poleProjection_eq_zero.mp
-      rw [map_sub, map_sub, poleProjection_monomial_image, ← heq, sub_self]
-  · rintro ⟨f, ⟨hf,hfj⟩,haf⟩
-    have hg : monomial j - f ∈ poleLattice (j - 1) := by
-      intro i hi
-      by_cases hij : i = j
-      · subst i; simp [HahnSeries.coeff_sub, hfj]
-      · simp [HahnSeries.coeff_sub, coeff_monomial, hij, hf i (by omega)]
-    refine ⟨a (monomial j - f), ⟨monomial j - f, hg, rfl⟩, ?_⟩
-    rw [map_sub, map_sub, poleProjection_monomial_image,
-      poleProjection_eq_zero.mpr haf, sub_zero]
-
-open WithZero
-
-lemma mem_poleLattice_iff_valuation (J : ℤ) (f : LaurentSeries k) :
-    f ∈ poleLattice J ↔ Valued.v f ≤ WithZero.exp J := by
-  rw [show J = -(-J) by omega, LaurentSeries.valuation_le_iff_coeff_lt_eq_zero]
-  constructor
-  · intro hf n hn
-    simpa using hf (-n) (by omega)
-  · intro hf i hi
-    exact hf (-i) (by omega)
-
-lemma poleLattice_eq_ball (J : ℤ) :
-    (poleLattice (k := k) J : Set (LaurentSeries k)) =
-      {f | (Valued.v : Valuation (LaurentSeries k) ℤᵐ⁰).restrict f ≤
-        (Valued.v : Valuation (LaurentSeries k) ℤᵐ⁰).restrict (monomial J)} := by
-  ext f
-  change f ∈ poleLattice J ↔ (Valued.v : Valuation (LaurentSeries k) ℤᵐ⁰).restrict f ≤
-    (Valued.v : Valuation (LaurentSeries k) ℤᵐ⁰).restrict (monomial J)
-  rw [Valuation.restrict_le_iff, mem_poleLattice_iff_valuation]
-  simp [monomial, LaurentSeries.valuation_single_zpow]
-
-lemma isClosed_poleLattice (J : ℤ) : IsClosed (poleLattice (k := k) J : Set (LaurentSeries k)) := by
-  rw [poleLattice_eq_ball]
-  exact Valued.isClosed_closedBall _ _
-
-lemma isOpen_poleLattice (J : ℤ) : IsOpen (poleLattice (k := k) J : Set (LaurentSeries k)) := by
-  rw [poleLattice_eq_ball]
-  apply Valued.isOpen_closedBall
-  rw [ne_eq, Valuation.restrict_eq_zero_iff]
-  simp [monomial, LaurentSeries.valuation_single_zpow]
-
-lemma poleLattice_nhds_basis :
-    (𝓝 (0 : LaurentSeries k)).HasBasis (fun _ : ℤ => True)
-      (fun J => (poleLattice (k := k) J : Set (LaurentSeries k))) := by
-  let v : Valuation (LaurentSeries k) ℤᵐ⁰ := Valued.v
-  apply (Valued.hasBasis_nhds_zero (LaurentSeries k) ℤᵐ⁰).to_hasBasis
-  · intro γ _
-    let r : ℤᵐ⁰ := MonoidWithZeroHom.ValueGroup₀.embedding γ.val
-    have hr : r ≠ 0 := (Units.map
-      (MonoidWithZeroHom.ValueGroup₀.embedding (f := .ofClass v)).toMonoidHom γ).ne_zero
-    refine ⟨WithZero.log r - 1, trivial, ?_⟩
-    intro f hf
-    rw [Set.mem_ofPred_eq, Valuation.restrict_lt_iff_lt_embedding]
-    apply (mem_poleLattice_iff_valuation _ f).mp hf |>.trans_lt
-    change WithZero.exp (WithZero.log r - 1) < r
-    calc
-      _ < WithZero.exp (WithZero.log r) := WithZero.exp_lt_exp.mpr (by omega)
-      _ = r := WithZero.exp_log hr
-  · intro J _
-    have hn : v.restrict (monomial J) ≠ 0 := by
-      rw [ne_eq, Valuation.restrict_eq_zero_iff]
-      simp [v, monomial, LaurentSeries.valuation_single_zpow]
-    refine ⟨Units.mk0 _ hn, trivial, ?_⟩
-    intro f hf
-    rw [poleLattice_eq_ball]
-    change v.restrict f < v.restrict (monomial J) at hf
-    exact hf.le
-
-lemma continuous_of_preservesPoles (a : LaurentSeries k →ₗ[k] LaurentSeries k)
-    (ha : PreservesPoles a) : Continuous a := by
-  apply continuous_of_continuousAt_zero a
-  simp only [ContinuousAt, map_zero]
-  rw [poleLattice_nhds_basis.tendsto_iff poleLattice_nhds_basis]
-  exact fun J _ => ⟨J, trivial, ha J⟩
-
-lemma continuous_of_bounded_coefficients [UniformSpace k] [DiscreteTopology k]
-    {X : Type*} [TopologicalSpace X] (g : X → LaurentSeries k) (J : ℤ)
-    (hg : ∀ x, g x ∈ poleLattice J)
-    (hcoeff : ∀ n : ℤ, Continuous (fun x => (g x).coeff n)) : Continuous g := by
-  rw [continuous_iff_continuousAt]
-  intro x
-  apply tendsto_sub_nhds_zero_iff.mp
-  rw [poleLattice_nhds_basis.tendsto_right_iff]
-  intro M _
-  have hev : ∀ᶠ y in 𝓝 x, ∀ i ∈ Finset.Ioc M J,
-      (g y).coeff (-i) = (g x).coeff (-i) := by
-    rw [Filter.eventually_all_finset]
-    intro i _
-    exact (hcoeff (-i)).continuousAt.eventually
-      ((isOpen_discrete {(g x).coeff (-i)}).mem_nhds rfl)
-  filter_upwards [hev] with y hy
-  intro i hi
-  rw [HahnSeries.coeff_sub, sub_eq_zero]
-  by_cases hij : i ≤ J
-  · exact hy i (Finset.mem_Ioc.mpr ⟨hi,hij⟩)
-  · rw [hg y i (by omega), hg x i (by omega)]
-
-def stageSeries (J : ℤ) (c : ℤ → k) : LaurentSeries k :=
-  HahnSeries.ofSuppBddBelow (fun n => if -J ≤ n then c n else 0)
-    ⟨-J, by
-      intro n hn
-      by_contra h
-      exact hn (ite_eq_right h)⟩
-
-@[simp] lemma coeff_stageSeries (J : ℤ) (c : ℤ → k) (n : ℤ) :
-    (stageSeries J c).coeff n = if -J ≤ n then c n else 0 := rfl
-
-lemma stageSeries_mem (J : ℤ) (c : ℤ → k) : stageSeries J c ∈ poleLattice J := by
-  intro i hi
-  rw [coeff_stageSeries, ite_eq_right (by omega)]
-
-lemma range_stageSeries (J : ℤ) :
-    Set.range (stageSeries (k := k) J) = (poleLattice (k := k) J : Set (LaurentSeries k)) := by
-  ext f
-  constructor
-  · rintro ⟨c,rfl⟩
-    exact stageSeries_mem J c
-  · intro hf
-    refine ⟨f.coeff, ?_⟩
-    apply HahnSeries.ext
-    funext n
-    rw [coeff_stageSeries]
-    split_ifs with hn
-    · rfl
-    · symm
-      simpa using hf (-n) (by omega)
-
-lemma continuous_stageSeries [UniformSpace k] [DiscreteTopology k] (J : ℤ) :
-    Continuous (stageSeries (k := k) J) := by
-  apply continuous_of_bounded_coefficients _ J (stageSeries_mem J)
-  intro n
-  simp only [coeff_stageSeries]
-  split_ifs
-  · exact continuous_apply n
-  · exact continuous_const
-
-lemma isCompact_poleLattice [Finite k] (J : ℤ) :
-    IsCompact (poleLattice (k := k) J : Set (LaurentSeries k)) := by
-  let : UniformSpace k := ⊥
-  rw [← range_stageSeries, ← Set.image_univ]
-  exact isCompact_univ.image (continuous_stageSeries J)
-
-lemma isCompact_leadingTerm [Finite k] (j : ℤ) :
-    IsCompact {f : LaurentSeries k | HasLeadingTerm f j} := by
-  let : UniformSpace k := ⊥
-  exact (isCompact_poleLattice j).inter_right
-    (isClosed_eq (LaurentSeries.uniformContinuous_coeff (-j)).continuous continuous_const)
-
-lemma isClosed_leadingTerm (j : ℤ) :
-    IsClosed {f : LaurentSeries k | HasLeadingTerm f j} := by
-  let : UniformSpace k := ⊥
-  exact (isClosed_poleLattice j).inter
-    (isClosed_eq (LaurentSeries.uniformContinuous_coeff (-j)).continuous continuous_const)
-
-lemma unmatched_iff_increments_zero (A : Matrix ℤ ℤ k)
-    (hA : RankMatching.Triangular A) (j : ℤ) :
-    (∀ l, ¬ RankMatching.Matched A l j) ↔ ∀ L, RankMatching.increment A L j = 0 := by
-  constructor
-  · intro h L
-    by_cases hLj : L ≤ j
-    · have hw : ∀ l, RankMatching.weight A l j = 0 := fun l =>
-        (RankMatching.weight_zero_or_one A hA l j).resolve_right (h l)
-      have hs := RankMatching.sum_interval_difference
-        (fun l => (RankMatching.increment A l j : ℤ)) L j (by omega)
-      simp_rw [← RankMatching.weight_eq_increment_sub A hA, hw] at hs
-      rw [RankMatching.increment_zero A hA (show j < j + 1 by omega)] at hs
-      simpa using hs.symm
-    · exact RankMatching.increment_zero A hA (by omega)
-  · intro h l hm
-    have hi := (RankMatching.matched_iff_increments A hA l j).mp hm
-    rw [h l] at hi
-    exact Nat.zero_ne_one hi.1
-
-theorem unmatched_iff_kernel [Finite k]
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (j : ℤ) :
-    (∀ l, ¬ RankMatching.Matched (matrixOf a) l j) ↔
-      ∃ f, a f = 0 ∧ HasLeadingTerm f j := by
-  rw [unmatched_iff_increments_zero _ (matrixOf_triangular a ha)]
-  constructor
-  · intro h
-    let S : ℤ → Set (LaurentSeries k) := fun L =>
-      {f | HasLeadingTerm f j ∧ a f ∈ poleLattice (L - 1)}
-    have hc : ∀ L, IsClosed (S L) := fun L =>
-      (isClosed_leadingTerm j).inter
-        ((isClosed_poleLattice (L - 1)).preimage (continuous_of_preservesPoles a ha))
-    have hk : ∀ L, IsCompact (S L) := fun L =>
-      (isCompact_leadingTerm j).of_isClosed_subset (hc L) (fun _ h => h.1)
-    have hn : ∀ L, (S L).Nonempty := fun L =>
-      (increment_zero_iff_finite_solution a ha L j).mp (h L)
-    have hd : Directed (fun s t : Set (LaurentSeries k) => s ⊇ t) S := by
-      intro L M
-      refine ⟨min L M, ?_, ?_⟩
-      · intro f hf
-        exact ⟨hf.1, poleLattice_mono (by omega) hf.2⟩
-      · intro f hf
-        exact ⟨hf.1, poleLattice_mono (by omega) hf.2⟩
-    obtain ⟨f,hf⟩ := IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed S hd hn hk hc
-    have hf' : ∀ L, f ∈ S L := Set.mem_iInter.mp hf
-    refine ⟨f, ?_, (hf' j).1⟩
-    apply HahnSeries.ext
-    funext n
-    simpa using (hf' (-n)).2 (-n) (by omega)
-  · rintro ⟨f,haf,hfj⟩ L
-    apply (increment_zero_iff_finite_solution a ha L j).mpr
-    exact ⟨f,hfj,haf ▸ (poleLattice (L - 1)).zero_mem⟩
-
-lemma rowRank_step (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A) (l J : ℤ) :
-    RankMatching.blockRank A l J = RankMatching.blockRank A (l + 1) J +
-      RankMatching.increment (RankMatching.reflected A) (-J) (-l) := by
-  have h := RankMatching.blockRank_step (RankMatching.reflected A)
-    (RankMatching.triangular_reflected A hA) (-J) (-l)
-  simpa only [RankMatching.blockRank_reflected, neg_neg, neg_sub,
-    Int.reduceNeg, sub_neg_eq_add, add_comm] using h
-
-lemma matchedRow_iff_unequal_ranks (A : Matrix ℤ ℤ k)
-    (hA : RankMatching.Triangular A) (l : ℤ) :
-    (∃ j, RankMatching.Matched A l j) ↔
-      ∃ J, RankMatching.blockRank A l J ≠ RankMatching.blockRank A (l + 1) J := by
-  have hn : (∀ j, ¬ RankMatching.Matched A l j) ↔
-      ∀ J, RankMatching.blockRank A l J = RankMatching.blockRank A (l + 1) J := by
-    have hh := unmatched_iff_increments_zero (RankMatching.reflected A)
-      (RankMatching.triangular_reflected A hA) (-l)
-    constructor
-    · intro h J
-      have hh' : ∀ r, ¬ RankMatching.Matched (RankMatching.reflected A) r (-l) := by
-        intro r hr
-        apply h (-r)
-        simpa only [RankMatching.Matched, RankMatching.weight_reflected, neg_neg] using hr
-      rw [rowRank_step A hA l J, hh.mp hh' (-J), add_zero]
-    · intro h j hm
-      have hh' : ∀ r, RankMatching.increment (RankMatching.reflected A) r (-l) = 0 := by
-        intro r
-        have hs := rowRank_step A hA l (-r)
-        rw [neg_neg, h (-r)] at hs
-        omega
-      apply hh.mpr hh' (-j)
-      simpa only [RankMatching.Matched, RankMatching.weight_reflected, neg_neg] using hm
-  simpa only [not_forall, not_not] using not_congr hn
-
-lemma exists_rowCut_kernel_of_unequal_ranks (A : Matrix ℤ ℤ k)
-    (hA : RankMatching.Triangular A) (l J : ℤ)
-    (hr : RankMatching.blockRank A l J ≠ RankMatching.blockRank A (l + 1) J) :
-    ∃ v ∈ RankMatching.columnSpace A l J,
-      RankMatching.rowCut (l + 1) v = 0 ∧ v l ≠ 0 := by
-  let F : RankMatching.columnSpace A l J →ₗ[k] RankMatching.columnSpace A (l + 1) J :=
-    (RankMatching.rowCut (l + 1)).restrict
-      (fun _ hv => RankMatching.rowCut_columnSpace A (by omega) J hv)
-  have hs : Function.Surjective F := by
-    intro v
-    have hm : (v : ℤ → k) ∈ (RankMatching.columnSpace A l J).map
-        (RankMatching.rowCut (l + 1)) := by
-      rw [RankMatching.rowCut_map_columnSpace A (by omega)]
-      exact v.property
-    obtain ⟨w,hw,he⟩ := Submodule.mem_map.mp hm
-    exact ⟨⟨w,hw⟩, Subtype.ext he⟩
-  have hni : ¬ Function.Injective F := by
-    intro hi
-    apply hr
-    have he := (LinearEquiv.ofBijective F ⟨hi,hs⟩).finrank_eq
-    simpa only [RankMatching.blockRank_eq_finrank A hA] using he
-  have hk : F.ker ≠ ⊥ := fun h => hni (LinearMap.ker_eq_bot.mp h)
-  obtain ⟨v,hv,hvz⟩ := F.ker.ne_bot_iff.mp hk
-  have hcut : RankMatching.rowCut (l + 1) (v : ℤ → k) = 0 :=
-    congrArg Subtype.val (show F v = 0 from hv)
-  refine ⟨v, v.property, hcut, ?_⟩
-  intro hvl
-  apply hvz
-  apply Subtype.ext
-  funext r
-  rcases lt_trichotomy r l with hlt | heq | hgt
-  · exact RankMatching.columnSpace_supported A l J v.property hlt
-  · simpa [heq] using hvl
-  · have he := congrFun hcut r
-    simpa [RankMatching.rowCut, hgt] using he
-
-theorem matched_row_iff_image
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (l : ℤ) :
-    (∃ j, RankMatching.Matched (matrixOf a) l j) ↔
-      ∃ f, HasLeadingPosition (a f) l := by
-  rw [matchedRow_iff_unequal_ranks _ (matrixOf_triangular a ha)]
-  constructor
-  · rintro ⟨J,hJ⟩
-    obtain ⟨v,hv,hcut,hvl⟩ := exists_rowCut_kernel_of_unequal_ranks
-      (matrixOf a) (matrixOf_triangular a ha) l J hJ
-    rw [columnSpace_eq_projectedImage a ha] at hv
-    obtain ⟨z,⟨f,hf,rfl⟩,rfl⟩ := hv
-    refine ⟨f, ?_, ?_⟩
-    · intro i hi
-      have he := congrFun hcut i
-      simpa [RankMatching.rowCut, poleProjection, hi,
-        show l ≤ i by omega] using he
-    · simpa [poleProjection] using hvl
-  · rintro ⟨f,hf,hfl⟩
-    obtain ⟨J,hJ⟩ := exists_poleLattice f
-    refine ⟨J, ?_⟩
-    intro heq
-    have hv : poleProjection l (a f) ∈ RankMatching.columnSpace (matrixOf a) l J := by
-      rw [columnSpace_eq_projectedImage a ha]
-      exact ⟨a f,⟨f,hJ,rfl⟩,rfl⟩
-    have hcut : RankMatching.rowCut (l + 1) (poleProjection l (a f)) = 0 := by
-      ext i
-      by_cases hi : l + 1 ≤ i
-      · simp [RankMatching.rowCut, poleProjection, show l ≤ i by omega,
-          hf i (by omega)]
-      · simp [RankMatching.rowCut, show ¬ l < i by omega]
-    have hi := RankMatching.rowCut_injOn_of_equal_ranks
-      (matrixOf a) (matrixOf_triangular a ha) l J heq
-    have he : poleProjection l (a f) = 0 := hi hv (Submodule.zero_mem _) (by simpa using hcut)
-    apply hfl
-    simpa [poleProjection] using congrFun he l
-
-def stageImage (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
-    Submodule k (LaurentSeries k) := (poleLattice J).map a
-
-lemma stageImage_mono (a : LaurentSeries k →ₗ[k] LaurentSeries k) : Monotone (stageImage a) := by
-  intro I J h
-  exact Submodule.map_mono (poleLattice_mono h)
-
-lemma stageImage_le_poleLattice (a : LaurentSeries k →ₗ[k] LaurentSeries k)
-    (ha : PreservesPoles a) (J : ℤ) : stageImage a J ≤ poleLattice J := by
-  rintro z ⟨f,hf,rfl⟩
-  exact ha J f hf
-
-lemma stageImage_step (a : LaurentSeries k →ₗ[k] LaurentSeries k) (j : ℤ) :
-    stageImage a j = stageImage a (j - 1) ⊔ Submodule.span k {a (monomial j)} := by
-  apply le_antisymm
-  · rintro z ⟨f,hf,rfl⟩
-    have ht : f - f.coeff (-j) • monomial j ∈ poleLattice (j - 1) := by
-      intro i hi
-      by_cases hij : i = j
-      · subst i; simp [HahnSeries.coeff_sub, HahnSeries.coeff_smul]
-      · simp [HahnSeries.coeff_sub, HahnSeries.coeff_smul, coeff_monomial, hij,
-          hf i (by omega)]
-    apply Submodule.mem_sup.mpr
-    refine ⟨a (f - f.coeff (-j) • monomial j), ⟨_,ht,rfl⟩,
-      f.coeff (-j) • a (monomial j),
-      Submodule.smul_mem _ _ (Submodule.subset_span (Set.mem_singleton _)), ?_⟩
-    simp
-  · apply sup_le (stageImage_mono a (by omega))
-    rw [Submodule.span_singleton_le_iff_mem]
-    exact ⟨monomial j, monomial_mem le_rfl, rfl⟩
-
-lemma monomial_image_mem_iff_unmatched [Finite k]
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (j : ℤ) :
-    a (monomial j) ∈ stageImage a (j - 1) ↔
-      ∀ l, ¬ RankMatching.Matched (matrixOf a) l j := by
-  rw [unmatched_iff_kernel a ha]
-  constructor
-  · rintro ⟨g,hg,he⟩
-    refine ⟨monomial j - g, by simp [he], ?_, ?_⟩
-    · intro i hi
-      simp [HahnSeries.coeff_sub, coeff_monomial, show i ≠ j by omega, hg i (by omega)]
-    · simp [HahnSeries.coeff_sub, hg j (by omega)]
-  · rintro ⟨f,haf,hf,hfj⟩
-    refine ⟨monomial j - f, ?_, by simp [haf]⟩
-    intro i hi
-    by_cases hij : i = j
-    · subst i; simp [HahnSeries.coeff_sub, hfj]
-    · simp [HahnSeries.coeff_sub, coeff_monomial, hij, hf i (by omega)]
-
-def imageModulo (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J N : ℤ) :
-    Submodule k (LaurentSeries k ⧸ stageImage a J) :=
-  (stageImage a N).map (stageImage a J).mkQ
-
-lemma imageModulo_mono (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
-    Monotone (imageModulo a J) := fun _ _ h => Submodule.map_mono (stageImage_mono a h)
-
-lemma imageModulo_zero (a : LaurentSeries k →ₗ[k] LaurentSeries k) {J N : ℤ} (h : N ≤ J) :
-    imageModulo a J N = ⊥ := by
-  apply le_antisymm _ bot_le
-  rw [imageModulo, Submodule.map_le_iff_le_comap, Submodule.comap_bot, Submodule.ker_mkQ]
-  exact stageImage_mono a h
-
-lemma imageModulo_step (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J n : ℤ) :
-    imageModulo a J n = imageModulo a J (n - 1) ⊔
-      Submodule.span k {(stageImage a J).mkQ (a (monomial n))} := by
-  rw [imageModulo, stageImage_step a n, Submodule.map_sup, Submodule.map_span, Set.image_singleton]
-  rfl
-
-lemma imageModulo_finite (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J N : ℤ) :
-    FiniteDimensional k (imageModulo a J N) := by
-  by_cases h : J ≤ N
-  · induction N, h using Int.leInduction with
-    | base => rw [imageModulo_zero a le_rfl]; infer_instance
-    | succ n hn ih =>
-      rw [imageModulo_step, add_sub_cancel_right]
-      let := ih
-      infer_instance
-  · rw [imageModulo_zero a (by omega)]
-    infer_instance
-
-lemma mem_imageModulo (a : LaurentSeries k →ₗ[k] LaurentSeries k) {J N : ℤ}
-    (h : J ≤ N) (f : LaurentSeries k) :
-    (stageImage a J).mkQ f ∈ imageModulo a J N ↔ f ∈ stageImage a N := by
-  change f ∈ ((stageImage a N).map (stageImage a J).mkQ).comap (stageImage a J).mkQ ↔ _
-  rw [Submodule.comap_map_eq, Submodule.ker_mkQ, sup_eq_left.mpr (stageImage_mono a h)]
-
-open scoped Classical
-
-lemma imageModulo_finrank_step [Finite k]
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) {J n : ℤ}
-    (h : J < n) :
-    Module.finrank k (imageModulo a J n) = Module.finrank k (imageModulo a J (n - 1)) +
-      if ∃ l, RankMatching.Matched (matrixOf a) l n then 1 else 0 := by
-  classical
-  let := imageModulo_finite a J (n - 1)
-  rw [imageModulo_step, RankMatching.finrank_sup_singleton,
-    mem_imageModulo a (by omega), monomial_image_mem_iff_unmatched a ha]
-  by_cases hm : ∃ l, RankMatching.Matched (matrixOf a) l n
-  · have hn : ¬ (∀ l, ¬ RankMatching.Matched (matrixOf a) l n) :=
-      fun hh => hh hm.choose hm.choose_spec
-    simp [hm, hn]
-  · simp [show ∀ l, ¬ RankMatching.Matched (matrixOf a) l n from fun l hl => hm ⟨l,hl⟩]
-
-lemma imageModulo_finrank [Finite k]
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (J N : ℤ) :
-    Module.finrank k (imageModulo a J N) =
-      ∑ j ∈ Finset.Ioc J N, if ∃ l, RankMatching.Matched (matrixOf a) l j then 1 else 0 := by
-  classical
-  by_cases h : J ≤ N
-  · induction N, h using Int.leInduction with
-    | base => rw [imageModulo_zero a (J := J) (N := J) le_rfl, finrank_bot]; simp
-    | succ n hn ih =>
-      have he : Finset.Ioc J (n + 1) = insert (n + 1) (Finset.Ioc J n) := by
-        ext i
-        simp only [Finset.mem_Ioc, Finset.mem_insert]
-        omega
-      rw [imageModulo_finrank_step a ha (by omega), add_sub_cancel_right, ih,
-        he, Finset.sum_insert (by simp)]
-      omega
-  · rw [imageModulo_zero a (by omega), finrank_bot, Finset.Ioc_eq_empty_of_le (by omega),
-      Finset.sum_empty]
-
-lemma matched_le (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A) {l j : ℤ}
-    (h : RankMatching.Matched A l j) : l ≤ j := by
-  by_contra hn
-  have hz := RankMatching.increment_zero A hA (lt_of_not_ge hn)
-  have ho := ((RankMatching.matched_iff_increments A hA l j).mp h).1
-  omega
-
-def columnMatches (A : Matrix ℤ ℤ k) (J N : ℤ) : Set (ℤ × ℤ) :=
-  {p | RankMatching.Matched A p.1 p.2 ∧ J < p.2 ∧ p.2 ≤ N}
-
-def crossingStageMatches (A : Matrix ℤ ℤ k) (J N : ℤ) : Set (ℤ × ℤ) :=
-  {p | RankMatching.Matched A p.1 p.2 ∧ p.1 ≤ J ∧ J < p.2 ∧ p.2 ≤ N}
-
-lemma columnMatches_snd_inj (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A)
-    (J N : ℤ) : Set.InjOn Prod.snd (columnMatches A J N) := by
-  rintro ⟨l,j⟩ hp ⟨l',j'⟩ hq he
-  change j = j' at he
-  subst j'
-  exact Prod.ext (RankMatching.column_unique A hA hp.1 hq.1) rfl
-
-lemma columnMatches_finite (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A)
-    (J N : ℤ) : (columnMatches A J N).Finite := by
-  exact Set.Finite.of_injOn (fun _ hp => hp.2) (columnMatches_snd_inj A hA J N)
-    (Set.finite_Ioc J N)
-
-lemma columnMatches_ncard (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A)
-    (J N : ℤ) : (columnMatches A J N).ncard =
-      ∑ j ∈ Finset.Ioc J N, if ∃ l, RankMatching.Matched A l j then 1 else 0 := by
-  classical
-  have he : Prod.snd '' columnMatches A J N =
-      ↑((Finset.Ioc J N).filter (fun j => ∃ l, RankMatching.Matched A l j)) := by
-    ext j
-    simp only [Set.mem_image, columnMatches, Set.mem_ofPred_eq, Finset.mem_coe,
-      Finset.mem_filter, Finset.mem_Ioc]
-    constructor
-    · rintro ⟨⟨l,j'⟩,⟨hm,hJ,hN⟩,rfl⟩
-      exact ⟨⟨hJ,hN⟩,l,hm⟩
-    · rintro ⟨⟨hJ,hN⟩,l,hm⟩
-      exact ⟨(l,j),⟨hm,hJ,hN⟩,rfl⟩
-  rw [← (columnMatches_snd_inj A hA J N).ncard_image, he,
-    Set.ncard_coe_finset, Finset.sum_boole]
-  rfl
-
-lemma columnMatches_partition (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A)
-    (J N : ℤ) :
-    (columnMatches A J N).ncard = (crossingStageMatches A J N).ncard +
-      RankMatching.blockRank A (J + 1) N := by
-  classical
-  let high := ((Finset.Icc (J + 1) N ×ˢ Finset.Icc (J + 1) N).filter
-    (fun p : ℤ × ℤ => RankMatching.Matched A p.1 p.2))
-  have he : columnMatches A J N = crossingStageMatches A J N ∪ (high : Set (ℤ × ℤ)) := by
-    ext p
-    simp only [columnMatches, crossingStageMatches, Set.mem_ofPred_eq, Set.mem_union,
-      high, Finset.mem_coe, Finset.mem_filter, Finset.mem_product, Finset.mem_Icc]
-    constructor
-    · rintro ⟨hm,hJ,hN⟩
-      by_cases hl : p.1 ≤ J
-      · exact Or.inl ⟨hm,hl,hJ,hN⟩
-      · have hh := matched_le A hA hm
-        exact Or.inr ⟨⟨⟨by omega,by omega⟩,⟨by omega,hN⟩⟩,hm⟩
-    · rintro (⟨hm,hl,hJ,hN⟩ | ⟨⟨hl,hj⟩,hm⟩)
-      · exact ⟨hm,hJ,hN⟩
-      · exact ⟨hm,by omega,hj.2⟩
-  have hd : Disjoint (crossingStageMatches A J N) (high : Set (ℤ × ℤ)) := by
-    apply Set.disjoint_left.mpr
-    intro p hp hq
-    simp only [high, Finset.mem_coe, Finset.mem_filter, Finset.mem_product,
-      Finset.mem_Icc] at hq
-    have hh := hp.2.1
-    omega
-  have hf : (crossingStageMatches A J N).Finite :=
-    (columnMatches_finite A hA J N).subset (fun _ h => ⟨h.1,h.2.2⟩)
-  rw [he, Set.ncard_union_eq hd hf high.finite_toSet, Set.ncard_coe_finset,
-    RankMatching.rank_eq_count A hA]
-  rfl
-
-lemma imageModulo_finrank_partition [Finite k]
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (J N : ℤ) :
-    Module.finrank k (imageModulo a J N) =
-      (crossingStageMatches (matrixOf a) J N).ncard + RankMatching.blockRank (matrixOf a) (J + 1) N := by
-  rw [imageModulo_finrank a ha, ← columnMatches_ncard _ (matrixOf_triangular a ha),
-    columnMatches_partition _ (matrixOf_triangular a ha)]
-
-lemma stageImage_le_ker_projection (a : LaurentSeries k →ₗ[k] LaurentSeries k)
-    (ha : PreservesPoles a) (J : ℤ) : stageImage a J ≤ (poleProjection (J + 1)).ker := by
-  intro f hf
-  change poleProjection (J + 1) f = 0
-  apply poleProjection_eq_zero.mpr
-  simpa using stageImage_le_poleLattice a ha J hf
-
-def quotientProjection (a : LaurentSeries k →ₗ[k] LaurentSeries k)
-    (ha : PreservesPoles a) (J : ℤ) :
-    (LaurentSeries k ⧸ stageImage a J) →ₗ[k] (ℤ → k) :=
-  (stageImage a J).liftQ (poleProjection (J + 1)) (stageImage_le_ker_projection a ha J)
-
-@[simp] lemma quotientProjection_mkQ (a : LaurentSeries k →ₗ[k] LaurentSeries k)
-    (ha : PreservesPoles a) (J : ℤ) (f : LaurentSeries k) :
-    quotientProjection a ha J ((stageImage a J).mkQ f) = poleProjection (J + 1) f := rfl
-
-def crossingStage (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J N : ℤ) :
-    Submodule k (LaurentSeries k ⧸ stageImage a J) :=
-  (stageImage a N ⊓ poleLattice J).map (stageImage a J).mkQ
-
-lemma crossingStage_mono (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
-    Monotone (crossingStage a J) := fun _ _ h =>
-  Submodule.map_mono (inf_le_inf_right _ (stageImage_mono a h))
-
-lemma crossingStage_finite (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J N : ℤ) :
-    FiniteDimensional k (crossingStage a J N) := by
-  let := imageModulo_finite a J N
-  exact Submodule.finiteDimensional_of_le (show crossingStage a J N ≤ imageModulo a J N
-    from Submodule.map_mono inf_le_left)
-
-lemma crossingStage_eq_inf (a : LaurentSeries k →ₗ[k] LaurentSeries k)
-    (ha : PreservesPoles a) (J N : ℤ) :
-    crossingStage a J N = imageModulo a J N ⊓ (quotientProjection a ha J).ker := by
-  ext z
-  constructor
-  · rintro ⟨f,⟨hf,hJ⟩,rfl⟩
-    exact ⟨⟨f,hf,rfl⟩, poleProjection_eq_zero.mpr (by simpa using hJ)⟩
-  · rintro ⟨⟨f,hf,rfl⟩,hz⟩
-    refine ⟨f,⟨hf,?_⟩,rfl⟩
-    have he : poleProjection (J + 1) f = 0 := hz
-    simpa using poleProjection_eq_zero.mp he
-
-lemma quotientProjection_stage_range (a : LaurentSeries k →ₗ[k] LaurentSeries k)
-    (ha : PreservesPoles a) (J N : ℤ) :
-    ((quotientProjection a ha J).domRestrict (imageModulo a J N)).range =
-      RankMatching.columnSpace (matrixOf a) (J + 1) N := by
-  rw [LinearMap.range_domRestrict, columnSpace_eq_projectedImage a ha]
-  ext v
-  constructor
-  · rintro ⟨z,⟨f,hf,rfl⟩,rfl⟩
-    exact ⟨f,hf,rfl⟩
-  · rintro ⟨f,hf,rfl⟩
-    exact ⟨(stageImage a J).mkQ f, ⟨f,hf,rfl⟩,rfl⟩
-
-lemma crossingStage_finrank [Finite k]
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (J N : ℤ) :
-    Module.finrank k (crossingStage a J N) = (crossingStageMatches (matrixOf a) J N).ncard := by
-  let := imageModulo_finite a J N
-  let f := (quotientProjection a ha J).domRestrict (imageModulo a J N)
-  have hr := LinearMap.finrank_range_add_finrank_ker f
-  have hk : Module.finrank k f.ker = Module.finrank k (crossingStage a J N) := by
-    rw [← Submodule.finrank_map_subtype_eq (imageModulo a J N) f.ker,
-      show f.ker = ((quotientProjection a ha J).ker).comap (imageModulo a J N).subtype
-        from LinearMap.ker_domRestrict _ _,
-      Submodule.map_comap_subtype, ← crossingStage_eq_inf a ha]
-  have hi : Module.finrank k f.range = RankMatching.blockRank (matrixOf a) (J + 1) N := by
-    rw [quotientProjection_stage_range, ← RankMatching.blockRank_eq_finrank _ (matrixOf_triangular a ha)]
-  rw [hk,hi,imageModulo_finrank_partition a ha] at hr
-  omega
-
-def crossingSpace (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
-    Submodule k (LaurentSeries k ⧸ stageImage a J) :=
-  (a.range ⊓ poleLattice J).map (stageImage a J).mkQ
-
-lemma crossingStage_le_space (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J N : ℤ) :
-    crossingStage a J N ≤ crossingSpace a J := by
-  apply Submodule.map_mono
-  apply inf_le_inf_right
-  rintro f ⟨g,hg,rfl⟩
-  exact ⟨g,rfl⟩
-
-lemma crossingSpace_stabilizes [Finite k]
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a)
-    (J : ℤ) (hc : (crossing a J).Finite) :
-    ∃ N : ℤ, J ≤ N ∧ crossingSpace a J = crossingStage a J N ∧
-      crossingStageMatches (matrixOf a) J N = crossing a J := by
-  obtain ⟨b,hb⟩ := (hc.image Prod.snd).bddAbove
-  let N := max J b
-  have he (M : ℤ) (hM : N ≤ M) :
-      crossingStageMatches (matrixOf a) J M = crossing a J := by
-    ext p
-    constructor
-    · intro hp
-      exact ⟨hp.1,hp.2.1,hp.2.2.1⟩
-    · intro hp
-      exact ⟨hp.1,hp.2.1,hp.2.2,le_trans (hb ⟨p,hp,rfl⟩) (le_trans (le_max_right J b) hM)⟩
-  refine ⟨N, le_max_left _ _, ?_, he N le_rfl⟩
-  apply le_antisymm _ (crossingStage_le_space a J N)
-  rintro z ⟨f,⟨⟨x,hx⟩,hf⟩,rfl⟩
-  obtain ⟨M,hM⟩ := exists_poleLattice x
-  have heq : crossingStage a J N = crossingStage a J (max N M) := by
-    let := crossingStage_finite a J (max N M)
-    apply Submodule.eq_of_le_of_finrank_eq (crossingStage_mono a J (le_max_left _ _))
-    rw [crossingStage_finrank a ha, crossingStage_finrank a ha,
-      he N le_rfl, he (max N M) (le_max_left _ _)]
-  rw [heq]
-  exact ⟨f,⟨⟨x,poleLattice_mono (le_max_right N M) hM,hx⟩,hf⟩,rfl⟩
-
-def crossingQuotientEquiv (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
-    crossingQuotient a J ≃ₗ[k] crossingSpace a J := by
-  let I := a.range ⊓ poleLattice (k := k) J
-  let f := (stageImage a J).mkQ.domRestrict I
-  have hk : f.ker = (stageImage a J).comap I.subtype := by
-    rw [LinearMap.ker_domRestrict, Submodule.ker_mkQ]
-  exact (Submodule.quotEquivOfEq _ _ hk.symm).trans
-    (f.quotKerEquivRange.trans (LinearEquiv.ofEq _ _ (LinearMap.range_domRestrict _ _)))
-
-theorem crossing_dimension [Finite k]
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a)
-    (hc : ∀ J, (crossing a J).Finite) (J : ℤ) :
-    FiniteDimensional k (crossingQuotient a J) ∧
-      Module.finrank k (crossingQuotient a J) = (crossing a J).ncard := by
-  obtain ⟨N,hN,he,hcN⟩ := crossingSpace_stabilizes a ha J (hc J)
-  have : FiniteDimensional k (crossingSpace a J) := by
-    rw [he]
-    exact crossingStage_finite a J N
-  refine ⟨(crossingQuotientEquiv a J).symm.finiteDimensional, ?_⟩
-  rw [(crossingQuotientEquiv a J).finrank_eq, he, crossingStage_finrank a ha, hcN]
-
-lemma range_inf_stabilizes [Finite k]
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a)
-    (J : ℤ) (hc : (crossing a J).Finite) :
-    ∃ N : ℤ, a.range ⊓ poleLattice J = stageImage a N ⊓ poleLattice J := by
-  obtain ⟨N,hN,he,_⟩ := crossingSpace_stabilizes a ha J hc
-  refine ⟨N,le_antisymm ?_ ?_⟩
-  · intro f hf
-    have hq : (stageImage a J).mkQ f ∈ crossingStage a J N := by
-      rw [← he]
-      exact ⟨f,hf,rfl⟩
-    have hi : (stageImage a J).mkQ f ∈ imageModulo a J N :=
-      (Submodule.map_mono (show stageImage a N ⊓ poleLattice J ≤ stageImage a N
-        from inf_le_left)) hq
-    exact ⟨(mem_imageModulo a hN f).mp hi,hf.2⟩
-  · rintro f ⟨⟨x,hx,rfl⟩,hf⟩
-    exact ⟨⟨x,rfl⟩,hf⟩
-
-lemma isCompact_range_inter_poleLattice [Finite k]
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a)
-    (J : ℤ) (hc : (crossing a J).Finite) :
-    IsCompact ((Set.range a) ∩ (poleLattice (k := k) J : Set (LaurentSeries k))) := by
-  obtain ⟨N,he⟩ := range_inf_stabilizes a ha J hc
-  change IsCompact (↑(a.range ⊓ poleLattice J) : Set (LaurentSeries k))
-  rw [he]
-  change IsCompact ((a '' (poleLattice (k := k) N : Set (LaurentSeries k))) ∩
-    (poleLattice (k := k) J : Set (LaurentSeries k)))
-  exact ((isCompact_poleLattice N).image (continuous_of_preservesPoles a ha)).inter_right
-    (isClosed_poleLattice J)
-
-theorem closed_image [Finite k]
-    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a)
-    (hc : ∀ J, (crossing a J).Finite) :
-    IsClosed (Set.range a) := by
-  rw [← isOpen_compl_iff, isOpen_iff_forall_mem_open]
-  intro f hf
-  obtain ⟨J,hJ⟩ := exists_poleLattice f
-  refine ⟨(poleLattice (k := k) J : Set (LaurentSeries k)) ∩
-    ((Set.range a) ∩ (poleLattice (k := k) J : Set (LaurentSeries k)))ᶜ, ?_, ?_, ?_⟩
-  · intro g hg hga
-    exact hg.2 ⟨hga,hg.1⟩
-  · exact (isOpen_poleLattice J).inter
-      (isCompact_range_inter_poleLattice a ha J (hc J)).isClosed.isOpen_compl
-  · exact ⟨hJ,fun h => hf h.1⟩
-
-end HahnWilson.PivotTopology
-
-end
-
-end OAI
+import OAI.LinearAlgebra.RankMatching.Basic
+
+namespace OAI
+
+noncomputable section
+open scoped BigOperators nonZeroDivisors
+open LinearMap Submodule
+open CategoryTheory CategoryTheory.Limits HomologicalComplex
+
+namespace HahnWilson.PivotTopology
+open HahnSeries
+open scoped BigOperators Topology
+variable {k : Type*} [Field k]
+
+def poleLattice (J : ℤ) : Submodule k (LaurentSeries k) where
+  carrier := {f | ∀ i : ℤ, J < i → f.coeff (-i) = 0}
+  zero_mem' := by simp
+  add_mem' hf hg i hi := by simp [hf i hi, hg i hi]
+  smul_mem' c f hf i hi := by simp [hf i hi]
+
+def monomial (j : ℤ) : LaurentSeries k := HahnSeries.single (-j) 1
+
+def PreservesPoles (a : LaurentSeries k →ₗ[k] LaurentSeries k) : Prop :=
+  ∀ J, ∀ f ∈ poleLattice (k := k) J, a f ∈ poleLattice (k := k) J
+
+def matrixOf (a : LaurentSeries k →ₗ[k] LaurentSeries k) : Matrix ℤ ℤ k :=
+  fun l j => (a (monomial j)).coeff (-l)
+
+def HasLeadingTerm (f : LaurentSeries k) (j : ℤ) : Prop :=
+  f ∈ poleLattice (k := k) j ∧ f.coeff (-j) = 1
+
+def HasLeadingPosition (f : LaurentSeries k) (j : ℤ) : Prop :=
+  f ∈ poleLattice (k := k) j ∧ f.coeff (-j) ≠ 0
+
+def crossing (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) : Set (ℤ × ℤ) :=
+  {p | RankMatching.Matched (matrixOf a) p.1 p.2 ∧ p.1 ≤ J ∧ J < p.2}
+
+def crossingQuotient (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :=
+  ↥(LinearMap.range a ⊓ poleLattice (k := k) J) ⧸
+    Submodule.comap (LinearMap.range a ⊓ poleLattice (k := k) J).subtype
+      ((poleLattice (k := k) J).map a)
+
+instance crossingQuotient_addCommGroup (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
+    AddCommGroup (crossingQuotient a J) := by
+  unfold crossingQuotient
+  infer_instance
+
+instance crossingQuotient_module (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
+    Module k (crossingQuotient a J) := by
+  unfold crossingQuotient
+  infer_instance
+
+@[simp] lemma mem_poleLattice (J : ℤ) (f : LaurentSeries k) :
+    f ∈ poleLattice J ↔ ∀ i : ℤ, J < i → f.coeff (-i) = 0 := Iff.rfl
+
+lemma poleLattice_mono {I J : ℤ} (h : I ≤ J) :
+    poleLattice (k := k) I ≤ poleLattice J := by
+  intro f hf i hi
+  exact hf i (lt_of_le_of_lt h hi)
+
+@[simp] lemma coeff_monomial (j i : ℤ) :
+    (monomial (k := k) j).coeff (-i) = if i = j then 1 else 0 := by
+  simp [monomial, HahnSeries.coeff_single]
+
+lemma monomial_mem {j J : ℤ} (h : j ≤ J) : monomial (k := k) j ∈ poleLattice J := by
+  intro i hi
+  simp [show i ≠ j by omega]
+
+lemma exists_poleLattice (f : LaurentSeries k) : ∃ J, f ∈ poleLattice J := by
+  refine ⟨-f.order, ?_⟩
+  intro i hi
+  exact HahnSeries.coeff_eq_zero_of_lt_order (by omega)
+
+lemma matrixOf_triangular (a : LaurentSeries k →ₗ[k] LaurentSeries k)
+    (ha : PreservesPoles a) : RankMatching.Triangular (matrixOf a) := by
+  intro l j h
+  exact ha j (monomial j) (monomial_mem le_rfl) l h
+
+def truncate (f : LaurentSeries k) (L J : ℤ) : LaurentSeries k :=
+  ∑ c ∈ Finset.Icc L J, f.coeff (-c) • monomial c
+
+lemma coeff_truncate (f : LaurentSeries k) (L J i : ℤ) :
+    (truncate f L J).coeff (-i) = if L ≤ i ∧ i ≤ J then f.coeff (-i) else 0 := by
+  classical
+  simp [truncate, HahnSeries.coeff_sum, HahnSeries.coeff_smul, smul_eq_mul,
+    mul_ite]
+
+lemma truncate_mem (f : LaurentSeries k) (L J : ℤ) : truncate f L J ∈ poleLattice J := by
+  intro i hi
+  rw [coeff_truncate, ite_eq_right (by omega)]
+
+lemma sub_truncate_mem {f : LaurentSeries k} {L J : ℤ} (hf : f ∈ poleLattice J) :
+    f - truncate f L J ∈ poleLattice (L - 1) := by
+  intro i hi
+  rw [HahnSeries.coeff_sub, coeff_truncate]
+  by_cases hij : i ≤ J
+  · rw [ite_eq_left (by omega), sub_self]
+  · rw [ite_eq_right (by omega), sub_zero, hf i (by omega)]
+
+def poleProjection (L : ℤ) : LaurentSeries k →ₗ[k] (ℤ → k) where
+  toFun f i := if L ≤ i then f.coeff (-i) else 0
+  map_add' f g := by ext i; by_cases hi : L ≤ i <;> simp [hi]
+  map_smul' c f := by ext i; by_cases hi : L ≤ i <;> simp [hi]
+
+lemma poleProjection_eq_zero {L : ℤ} {f : LaurentSeries k} :
+    poleProjection L f = 0 ↔ f ∈ poleLattice (L - 1) := by
+  constructor
+  · intro h i hi
+    have := congrFun h i
+    simpa [poleProjection, show L ≤ i by omega] using this
+  · intro h
+    ext i
+    by_cases hi : L ≤ i
+    · simp [poleProjection, hi, h i (by omega)]
+    · simp [poleProjection, hi]
+
+@[simp] lemma poleProjection_monomial_image
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (L j : ℤ) :
+    poleProjection L (a (monomial j)) = RankMatching.column (matrixOf a) L j := rfl
+
+lemma poleProjection_truncate_image (a : LaurentSeries k →ₗ[k] LaurentSeries k)
+    (ha : PreservesPoles a) {f : LaurentSeries k} {L J : ℤ}
+    (hf : f ∈ poleLattice J) :
+    poleProjection L (a (truncate f L J)) = poleProjection L (a f) := by
+  have h := (poleProjection_eq_zero).mpr (ha (L - 1) _ (sub_truncate_mem hf))
+  rw [map_sub, map_sub, sub_eq_zero] at h
+  exact h.symm
+
+lemma columnSpace_eq_projectedImage (a : LaurentSeries k →ₗ[k] LaurentSeries k)
+    (ha : PreservesPoles a) (L J : ℤ) :
+    RankMatching.columnSpace (matrixOf a) L J =
+      ((poleLattice J).map a).map (poleProjection L) := by
+  classical
+  apply le_antisymm
+  · apply Submodule.span_le.mpr
+    rintro _ ⟨c, hc, rfl⟩
+    exact ⟨a (monomial c), ⟨monomial c, monomial_mem hc, rfl⟩, rfl⟩
+  · rintro y ⟨z, ⟨f,hf,rfl⟩,rfl⟩
+    rw [← poleProjection_truncate_image a ha hf]
+    simp only [truncate, map_sum, map_smul]
+    exact Submodule.sum_mem _ fun c hc =>
+      Submodule.smul_mem _ _ (Submodule.subset_span
+        ⟨c, (Finset.mem_Icc.mp hc).2, rfl⟩)
+
+lemma increment_zero_iff_finite_solution
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (L j : ℤ) :
+    RankMatching.increment (matrixOf a) L j = 0 ↔
+      ∃ f, HasLeadingTerm f j ∧ a f ∈ poleLattice (L - 1) := by
+  rw [RankMatching.increment_eq_zero_iff, columnSpace_eq_projectedImage a ha]
+  constructor
+  · rintro ⟨z, ⟨g,hg,rfl⟩,heq⟩
+    refine ⟨monomial j - g, ⟨?_, ?_⟩, ?_⟩
+    · intro i hi
+      simp only [HahnSeries.coeff_sub, coeff_monomial,
+        ite_eq_right (show i ≠ j by omega), hg i (by omega), sub_self]
+    · simp [HahnSeries.coeff_sub, hg j (by omega)]
+    · apply poleProjection_eq_zero.mp
+      rw [map_sub, map_sub, poleProjection_monomial_image, ← heq, sub_self]
+  · rintro ⟨f, ⟨hf,hfj⟩,haf⟩
+    have hg : monomial j - f ∈ poleLattice (j - 1) := by
+      intro i hi
+      by_cases hij : i = j
+      · subst i; simp [HahnSeries.coeff_sub, hfj]
+      · simp [HahnSeries.coeff_sub, coeff_monomial, hij, hf i (by omega)]
+    refine ⟨a (monomial j - f), ⟨monomial j - f, hg, rfl⟩, ?_⟩
+    rw [map_sub, map_sub, poleProjection_monomial_image,
+      poleProjection_eq_zero.mpr haf, sub_zero]
+
+open WithZero
+
+lemma mem_poleLattice_iff_valuation (J : ℤ) (f : LaurentSeries k) :
+    f ∈ poleLattice J ↔ Valued.v f ≤ WithZero.exp J := by
+  rw [show J = -(-J) by omega, LaurentSeries.valuation_le_iff_coeff_lt_eq_zero]
+  constructor
+  · intro hf n hn
+    simpa using hf (-n) (by omega)
+  · intro hf i hi
+    exact hf (-i) (by omega)
+
+lemma poleLattice_eq_ball (J : ℤ) :
+    (poleLattice (k := k) J : Set (LaurentSeries k)) =
+      {f | (Valued.v : Valuation (LaurentSeries k) ℤᵐ⁰).restrict f ≤
+        (Valued.v : Valuation (LaurentSeries k) ℤᵐ⁰).restrict (monomial J)} := by
+  ext f
+  change f ∈ poleLattice J ↔ (Valued.v : Valuation (LaurentSeries k) ℤᵐ⁰).restrict f ≤
+    (Valued.v : Valuation (LaurentSeries k) ℤᵐ⁰).restrict (monomial J)
+  rw [Valuation.restrict_le_iff, mem_poleLattice_iff_valuation]
+  simp [monomial, LaurentSeries.valuation_single_zpow]
+
+lemma isClosed_poleLattice (J : ℤ) : IsClosed (poleLattice (k := k) J : Set (LaurentSeries k)) := by
+  rw [poleLattice_eq_ball]
+  exact Valued.isClosed_closedBall _ _
+
+lemma isOpen_poleLattice (J : ℤ) : IsOpen (poleLattice (k := k) J : Set (LaurentSeries k)) := by
+  rw [poleLattice_eq_ball]
+  apply Valued.isOpen_closedBall
+  rw [ne_eq, Valuation.restrict_eq_zero_iff]
+  simp [monomial, LaurentSeries.valuation_single_zpow]
+
+lemma poleLattice_nhds_basis :
+    (𝓝 (0 : LaurentSeries k)).HasBasis (fun _ : ℤ => True)
+      (fun J => (poleLattice (k := k) J : Set (LaurentSeries k))) := by
+  let v : Valuation (LaurentSeries k) ℤᵐ⁰ := Valued.v
+  apply (Valued.hasBasis_nhds_zero (LaurentSeries k) ℤᵐ⁰).to_hasBasis
+  · intro γ _
+    let r : ℤᵐ⁰ := MonoidWithZeroHom.ValueGroup₀.embedding γ.val
+    have hr : r ≠ 0 := (Units.map
+      (MonoidWithZeroHom.ValueGroup₀.embedding (f := .ofClass v)).toMonoidHom γ).ne_zero
+    refine ⟨WithZero.log r - 1, trivial, ?_⟩
+    intro f hf
+    rw [Set.mem_ofPred_eq, Valuation.restrict_lt_iff_lt_embedding]
+    apply (mem_poleLattice_iff_valuation _ f).mp hf |>.trans_lt
+    change WithZero.exp (WithZero.log r - 1) < r
+    calc
+      _ < WithZero.exp (WithZero.log r) := WithZero.exp_lt_exp.mpr (by omega)
+      _ = r := WithZero.exp_log hr
+  · intro J _
+    have hn : v.restrict (monomial J) ≠ 0 := by
+      rw [ne_eq, Valuation.restrict_eq_zero_iff]
+      simp [v, monomial, LaurentSeries.valuation_single_zpow]
+    refine ⟨Units.mk0 _ hn, trivial, ?_⟩
+    intro f hf
+    rw [poleLattice_eq_ball]
+    change v.restrict f < v.restrict (monomial J) at hf
+    exact hf.le
+
+lemma continuous_of_preservesPoles (a : LaurentSeries k →ₗ[k] LaurentSeries k)
+    (ha : PreservesPoles a) : Continuous a := by
+  apply continuous_of_continuousAt_zero a
+  simp only [ContinuousAt, map_zero]
+  rw [poleLattice_nhds_basis.tendsto_iff poleLattice_nhds_basis]
+  exact fun J _ => ⟨J, trivial, ha J⟩
+
+lemma continuous_of_bounded_coefficients [UniformSpace k] [DiscreteTopology k]
+    {X : Type*} [TopologicalSpace X] (g : X → LaurentSeries k) (J : ℤ)
+    (hg : ∀ x, g x ∈ poleLattice J)
+    (hcoeff : ∀ n : ℤ, Continuous (fun x => (g x).coeff n)) : Continuous g := by
+  rw [continuous_iff_continuousAt]
+  intro x
+  apply tendsto_sub_nhds_zero_iff.mp
+  rw [poleLattice_nhds_basis.tendsto_right_iff]
+  intro M _
+  have hev : ∀ᶠ y in 𝓝 x, ∀ i ∈ Finset.Ioc M J,
+      (g y).coeff (-i) = (g x).coeff (-i) := by
+    rw [Filter.eventually_all_finset]
+    intro i _
+    exact (hcoeff (-i)).continuousAt.eventually
+      ((isOpen_discrete {(g x).coeff (-i)}).mem_nhds rfl)
+  filter_upwards [hev] with y hy
+  intro i hi
+  rw [HahnSeries.coeff_sub, sub_eq_zero]
+  by_cases hij : i ≤ J
+  · exact hy i (Finset.mem_Ioc.mpr ⟨hi,hij⟩)
+  · rw [hg y i (by omega), hg x i (by omega)]
+
+def stageSeries (J : ℤ) (c : ℤ → k) : LaurentSeries k :=
+  HahnSeries.ofSuppBddBelow (fun n => if -J ≤ n then c n else 0)
+    ⟨-J, by
+      intro n hn
+      by_contra h
+      exact hn (ite_eq_right h)⟩
+
+@[simp] lemma coeff_stageSeries (J : ℤ) (c : ℤ → k) (n : ℤ) :
+    (stageSeries J c).coeff n = if -J ≤ n then c n else 0 := rfl
+
+lemma stageSeries_mem (J : ℤ) (c : ℤ → k) : stageSeries J c ∈ poleLattice J := by
+  intro i hi
+  rw [coeff_stageSeries, ite_eq_right (by omega)]
+
+lemma range_stageSeries (J : ℤ) :
+    Set.range (stageSeries (k := k) J) = (poleLattice (k := k) J : Set (LaurentSeries k)) := by
+  ext f
+  constructor
+  · rintro ⟨c,rfl⟩
+    exact stageSeries_mem J c
+  · intro hf
+    refine ⟨f.coeff, ?_⟩
+    apply HahnSeries.ext
+    funext n
+    rw [coeff_stageSeries]
+    split_ifs with hn
+    · rfl
+    · symm
+      simpa using hf (-n) (by omega)
+
+lemma continuous_stageSeries [UniformSpace k] [DiscreteTopology k] (J : ℤ) :
+    Continuous (stageSeries (k := k) J) := by
+  apply continuous_of_bounded_coefficients _ J (stageSeries_mem J)
+  intro n
+  simp only [coeff_stageSeries]
+  split_ifs
+  · exact continuous_apply n
+  · exact continuous_const
+
+lemma isCompact_poleLattice [Finite k] (J : ℤ) :
+    IsCompact (poleLattice (k := k) J : Set (LaurentSeries k)) := by
+  let : UniformSpace k := ⊥
+  rw [← range_stageSeries, ← Set.image_univ]
+  exact isCompact_univ.image (continuous_stageSeries J)
+
+lemma isCompact_leadingTerm [Finite k] (j : ℤ) :
+    IsCompact {f : LaurentSeries k | HasLeadingTerm f j} := by
+  let : UniformSpace k := ⊥
+  exact (isCompact_poleLattice j).inter_right
+    (isClosed_eq (LaurentSeries.uniformContinuous_coeff (-j)).continuous continuous_const)
+
+lemma isClosed_leadingTerm (j : ℤ) :
+    IsClosed {f : LaurentSeries k | HasLeadingTerm f j} := by
+  let : UniformSpace k := ⊥
+  exact (isClosed_poleLattice j).inter
+    (isClosed_eq (LaurentSeries.uniformContinuous_coeff (-j)).continuous continuous_const)
+
+lemma unmatched_iff_increments_zero (A : Matrix ℤ ℤ k)
+    (hA : RankMatching.Triangular A) (j : ℤ) :
+    (∀ l, ¬ RankMatching.Matched A l j) ↔ ∀ L, RankMatching.increment A L j = 0 := by
+  constructor
+  · intro h L
+    by_cases hLj : L ≤ j
+    · have hw : ∀ l, RankMatching.weight A l j = 0 := fun l =>
+        (RankMatching.weight_zero_or_one A hA l j).resolve_right (h l)
+      have hs := RankMatching.sum_interval_difference
+        (fun l => (RankMatching.increment A l j : ℤ)) L j (by omega)
+      simp_rw [← RankMatching.weight_eq_increment_sub A hA, hw] at hs
+      rw [RankMatching.increment_zero A hA (show j < j + 1 by omega)] at hs
+      simpa using hs.symm
+    · exact RankMatching.increment_zero A hA (by omega)
+  · intro h l hm
+    have hi := (RankMatching.matched_iff_increments A hA l j).mp hm
+    rw [h l] at hi
+    exact Nat.zero_ne_one hi.1
+
+theorem unmatched_iff_kernel [Finite k]
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (j : ℤ) :
+    (∀ l, ¬ RankMatching.Matched (matrixOf a) l j) ↔
+      ∃ f, a f = 0 ∧ HasLeadingTerm f j := by
+  rw [unmatched_iff_increments_zero _ (matrixOf_triangular a ha)]
+  constructor
+  · intro h
+    let S : ℤ → Set (LaurentSeries k) := fun L =>
+      {f | HasLeadingTerm f j ∧ a f ∈ poleLattice (L - 1)}
+    have hc : ∀ L, IsClosed (S L) := fun L =>
+      (isClosed_leadingTerm j).inter
+        ((isClosed_poleLattice (L - 1)).preimage (continuous_of_preservesPoles a ha))
+    have hk : ∀ L, IsCompact (S L) := fun L =>
+      (isCompact_leadingTerm j).of_isClosed_subset (hc L) (fun _ h => h.1)
+    have hn : ∀ L, (S L).Nonempty := fun L =>
+      (increment_zero_iff_finite_solution a ha L j).mp (h L)
+    have hd : Directed (fun s t : Set (LaurentSeries k) => s ⊇ t) S := by
+      intro L M
+      refine ⟨min L M, ?_, ?_⟩
+      · intro f hf
+        exact ⟨hf.1, poleLattice_mono (by omega) hf.2⟩
+      · intro f hf
+        exact ⟨hf.1, poleLattice_mono (by omega) hf.2⟩
+    obtain ⟨f,hf⟩ := IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed S hd hn hk hc
+    have hf' : ∀ L, f ∈ S L := Set.mem_iInter.mp hf
+    refine ⟨f, ?_, (hf' j).1⟩
+    apply HahnSeries.ext
+    funext n
+    simpa using (hf' (-n)).2 (-n) (by omega)
+  · rintro ⟨f,haf,hfj⟩ L
+    apply (increment_zero_iff_finite_solution a ha L j).mpr
+    exact ⟨f,hfj,haf ▸ (poleLattice (L - 1)).zero_mem⟩
+
+lemma rowRank_step (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A) (l J : ℤ) :
+    RankMatching.blockRank A l J = RankMatching.blockRank A (l + 1) J +
+      RankMatching.increment (RankMatching.reflected A) (-J) (-l) := by
+  have h := RankMatching.blockRank_step (RankMatching.reflected A)
+    (RankMatching.triangular_reflected A hA) (-J) (-l)
+  simpa only [RankMatching.blockRank_reflected, neg_neg, neg_sub,
+    Int.reduceNeg, sub_neg_eq_add, add_comm] using h
+
+lemma matchedRow_iff_unequal_ranks (A : Matrix ℤ ℤ k)
+    (hA : RankMatching.Triangular A) (l : ℤ) :
+    (∃ j, RankMatching.Matched A l j) ↔
+      ∃ J, RankMatching.blockRank A l J ≠ RankMatching.blockRank A (l + 1) J := by
+  have hn : (∀ j, ¬ RankMatching.Matched A l j) ↔
+      ∀ J, RankMatching.blockRank A l J = RankMatching.blockRank A (l + 1) J := by
+    have hh := unmatched_iff_increments_zero (RankMatching.reflected A)
+      (RankMatching.triangular_reflected A hA) (-l)
+    constructor
+    · intro h J
+      have hh' : ∀ r, ¬ RankMatching.Matched (RankMatching.reflected A) r (-l) := by
+        intro r hr
+        apply h (-r)
+        simpa only [RankMatching.Matched, RankMatching.weight_reflected, neg_neg] using hr
+      rw [rowRank_step A hA l J, hh.mp hh' (-J), add_zero]
+    · intro h j hm
+      have hh' : ∀ r, RankMatching.increment (RankMatching.reflected A) r (-l) = 0 := by
+        intro r
+        have hs := rowRank_step A hA l (-r)
+        rw [neg_neg, h (-r)] at hs
+        omega
+      apply hh.mpr hh' (-j)
+      simpa only [RankMatching.Matched, RankMatching.weight_reflected, neg_neg] using hm
+  simpa only [not_forall, not_not] using not_congr hn
+
+lemma exists_rowCut_kernel_of_unequal_ranks (A : Matrix ℤ ℤ k)
+    (hA : RankMatching.Triangular A) (l J : ℤ)
+    (hr : RankMatching.blockRank A l J ≠ RankMatching.blockRank A (l + 1) J) :
+    ∃ v ∈ RankMatching.columnSpace A l J,
+      RankMatching.rowCut (l + 1) v = 0 ∧ v l ≠ 0 := by
+  let F : RankMatching.columnSpace A l J →ₗ[k] RankMatching.columnSpace A (l + 1) J :=
+    (RankMatching.rowCut (l + 1)).restrict
+      (fun _ hv => RankMatching.rowCut_columnSpace A (by omega) J hv)
+  have hs : Function.Surjective F := by
+    intro v
+    have hm : (v : ℤ → k) ∈ (RankMatching.columnSpace A l J).map
+        (RankMatching.rowCut (l + 1)) := by
+      rw [RankMatching.rowCut_map_columnSpace A (by omega)]
+      exact v.property
+    obtain ⟨w,hw,he⟩ := Submodule.mem_map.mp hm
+    exact ⟨⟨w,hw⟩, Subtype.ext he⟩
+  have hni : ¬ Function.Injective F := by
+    intro hi
+    apply hr
+    have he := (LinearEquiv.ofBijective F ⟨hi,hs⟩).finrank_eq
+    simpa only [RankMatching.blockRank_eq_finrank A hA] using he
+  have hk : F.ker ≠ ⊥ := fun h => hni (LinearMap.ker_eq_bot.mp h)
+  obtain ⟨v,hv,hvz⟩ := F.ker.ne_bot_iff.mp hk
+  have hcut : RankMatching.rowCut (l + 1) (v : ℤ → k) = 0 :=
+    congrArg Subtype.val (show F v = 0 from hv)
+  refine ⟨v, v.property, hcut, ?_⟩
+  intro hvl
+  apply hvz
+  apply Subtype.ext
+  funext r
+  rcases lt_trichotomy r l with hlt | heq | hgt
+  · exact RankMatching.columnSpace_supported A l J v.property hlt
+  · simpa [heq] using hvl
+  · have he := congrFun hcut r
+    simpa [RankMatching.rowCut, hgt] using he
+
+theorem matched_row_iff_image
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (l : ℤ) :
+    (∃ j, RankMatching.Matched (matrixOf a) l j) ↔
+      ∃ f, HasLeadingPosition (a f) l := by
+  rw [matchedRow_iff_unequal_ranks _ (matrixOf_triangular a ha)]
+  constructor
+  · rintro ⟨J,hJ⟩
+    obtain ⟨v,hv,hcut,hvl⟩ := exists_rowCut_kernel_of_unequal_ranks
+      (matrixOf a) (matrixOf_triangular a ha) l J hJ
+    rw [columnSpace_eq_projectedImage a ha] at hv
+    obtain ⟨z,⟨f,hf,rfl⟩,rfl⟩ := hv
+    refine ⟨f, ?_, ?_⟩
+    · intro i hi
+      have he := congrFun hcut i
+      simpa [RankMatching.rowCut, poleProjection, hi,
+        show l ≤ i by omega] using he
+    · simpa [poleProjection] using hvl
+  · rintro ⟨f,hf,hfl⟩
+    obtain ⟨J,hJ⟩ := exists_poleLattice f
+    refine ⟨J, ?_⟩
+    intro heq
+    have hv : poleProjection l (a f) ∈ RankMatching.columnSpace (matrixOf a) l J := by
+      rw [columnSpace_eq_projectedImage a ha]
+      exact ⟨a f,⟨f,hJ,rfl⟩,rfl⟩
+    have hcut : RankMatching.rowCut (l + 1) (poleProjection l (a f)) = 0 := by
+      ext i
+      by_cases hi : l + 1 ≤ i
+      · simp [RankMatching.rowCut, poleProjection, show l ≤ i by omega,
+          hf i (by omega)]
+      · simp [RankMatching.rowCut, show ¬ l < i by omega]
+    have hi := RankMatching.rowCut_injOn_of_equal_ranks
+      (matrixOf a) (matrixOf_triangular a ha) l J heq
+    have he : poleProjection l (a f) = 0 := hi hv (Submodule.zero_mem _) (by simpa using hcut)
+    apply hfl
+    simpa [poleProjection] using congrFun he l
+
+def stageImage (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
+    Submodule k (LaurentSeries k) := (poleLattice J).map a
+
+lemma stageImage_mono (a : LaurentSeries k →ₗ[k] LaurentSeries k) : Monotone (stageImage a) := by
+  intro I J h
+  exact Submodule.map_mono (poleLattice_mono h)
+
+lemma stageImage_le_poleLattice (a : LaurentSeries k →ₗ[k] LaurentSeries k)
+    (ha : PreservesPoles a) (J : ℤ) : stageImage a J ≤ poleLattice J := by
+  rintro z ⟨f,hf,rfl⟩
+  exact ha J f hf
+
+lemma stageImage_step (a : LaurentSeries k →ₗ[k] LaurentSeries k) (j : ℤ) :
+    stageImage a j = stageImage a (j - 1) ⊔ Submodule.span k {a (monomial j)} := by
+  apply le_antisymm
+  · rintro z ⟨f,hf,rfl⟩
+    have ht : f - f.coeff (-j) • monomial j ∈ poleLattice (j - 1) := by
+      intro i hi
+      by_cases hij : i = j
+      · subst i; simp [HahnSeries.coeff_sub, HahnSeries.coeff_smul]
+      · simp [HahnSeries.coeff_sub, HahnSeries.coeff_smul, coeff_monomial, hij,
+          hf i (by omega)]
+    apply Submodule.mem_sup.mpr
+    refine ⟨a (f - f.coeff (-j) • monomial j), ⟨_,ht,rfl⟩,
+      f.coeff (-j) • a (monomial j),
+      Submodule.smul_mem _ _ (Submodule.subset_span (Set.mem_singleton _)), ?_⟩
+    simp
+  · apply sup_le (stageImage_mono a (by omega))
+    rw [Submodule.span_singleton_le_iff_mem]
+    exact ⟨monomial j, monomial_mem le_rfl, rfl⟩
+
+lemma monomial_image_mem_iff_unmatched [Finite k]
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (j : ℤ) :
+    a (monomial j) ∈ stageImage a (j - 1) ↔
+      ∀ l, ¬ RankMatching.Matched (matrixOf a) l j := by
+  rw [unmatched_iff_kernel a ha]
+  constructor
+  · rintro ⟨g,hg,he⟩
+    refine ⟨monomial j - g, by simp [he], ?_, ?_⟩
+    · intro i hi
+      simp [HahnSeries.coeff_sub, coeff_monomial, show i ≠ j by omega, hg i (by omega)]
+    · simp [HahnSeries.coeff_sub, hg j (by omega)]
+  · rintro ⟨f,haf,hf,hfj⟩
+    refine ⟨monomial j - f, ?_, by simp [haf]⟩
+    intro i hi
+    by_cases hij : i = j
+    · subst i; simp [HahnSeries.coeff_sub, hfj]
+    · simp [HahnSeries.coeff_sub, coeff_monomial, hij, hf i (by omega)]
+
+def imageModulo (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J N : ℤ) :
+    Submodule k (LaurentSeries k ⧸ stageImage a J) :=
+  (stageImage a N).map (stageImage a J).mkQ
+
+lemma imageModulo_mono (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
+    Monotone (imageModulo a J) := fun _ _ h => Submodule.map_mono (stageImage_mono a h)
+
+lemma imageModulo_zero (a : LaurentSeries k →ₗ[k] LaurentSeries k) {J N : ℤ} (h : N ≤ J) :
+    imageModulo a J N = ⊥ := by
+  apply le_antisymm _ bot_le
+  rw [imageModulo, Submodule.map_le_iff_le_comap, Submodule.comap_bot, Submodule.ker_mkQ]
+  exact stageImage_mono a h
+
+lemma imageModulo_step (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J n : ℤ) :
+    imageModulo a J n = imageModulo a J (n - 1) ⊔
+      Submodule.span k {(stageImage a J).mkQ (a (monomial n))} := by
+  rw [imageModulo, stageImage_step a n, Submodule.map_sup, Submodule.map_span, Set.image_singleton]
+  rfl
+
+lemma imageModulo_finite (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J N : ℤ) :
+    FiniteDimensional k (imageModulo a J N) := by
+  by_cases h : J ≤ N
+  · induction N, h using Int.leInduction with
+    | base => rw [imageModulo_zero a le_rfl]; infer_instance
+    | succ n hn ih =>
+      rw [imageModulo_step, add_sub_cancel_right]
+      let := ih
+      infer_instance
+  · rw [imageModulo_zero a (by omega)]
+    infer_instance
+
+lemma mem_imageModulo (a : LaurentSeries k →ₗ[k] LaurentSeries k) {J N : ℤ}
+    (h : J ≤ N) (f : LaurentSeries k) :
+    (stageImage a J).mkQ f ∈ imageModulo a J N ↔ f ∈ stageImage a N := by
+  change f ∈ ((stageImage a N).map (stageImage a J).mkQ).comap (stageImage a J).mkQ ↔ _
+  rw [Submodule.comap_map_eq, Submodule.ker_mkQ, sup_eq_left.mpr (stageImage_mono a h)]
+
+open scoped Classical
+
+lemma imageModulo_finrank_step [Finite k]
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) {J n : ℤ}
+    (h : J < n) :
+    Module.finrank k (imageModulo a J n) = Module.finrank k (imageModulo a J (n - 1)) +
+      if ∃ l, RankMatching.Matched (matrixOf a) l n then 1 else 0 := by
+  classical
+  let := imageModulo_finite a J (n - 1)
+  rw [imageModulo_step, RankMatching.finrank_sup_singleton,
+    mem_imageModulo a (by omega), monomial_image_mem_iff_unmatched a ha]
+  by_cases hm : ∃ l, RankMatching.Matched (matrixOf a) l n
+  · have hn : ¬ (∀ l, ¬ RankMatching.Matched (matrixOf a) l n) :=
+      fun hh => hh hm.choose hm.choose_spec
+    simp [hm, hn]
+  · simp [show ∀ l, ¬ RankMatching.Matched (matrixOf a) l n from fun l hl => hm ⟨l,hl⟩]
+
+lemma imageModulo_finrank [Finite k]
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (J N : ℤ) :
+    Module.finrank k (imageModulo a J N) =
+      ∑ j ∈ Finset.Ioc J N, if ∃ l, RankMatching.Matched (matrixOf a) l j then 1 else 0 := by
+  classical
+  by_cases h : J ≤ N
+  · induction N, h using Int.leInduction with
+    | base => rw [imageModulo_zero a (J := J) (N := J) le_rfl, finrank_bot]; simp
+    | succ n hn ih =>
+      have he : Finset.Ioc J (n + 1) = insert (n + 1) (Finset.Ioc J n) := by
+        ext i
+        simp only [Finset.mem_Ioc, Finset.mem_insert]
+        omega
+      rw [imageModulo_finrank_step a ha (by omega), add_sub_cancel_right, ih,
+        he, Finset.sum_insert (by simp)]
+      omega
+  · rw [imageModulo_zero a (by omega), finrank_bot, Finset.Ioc_eq_empty_of_le (by omega),
+      Finset.sum_empty]
+
+lemma matched_le (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A) {l j : ℤ}
+    (h : RankMatching.Matched A l j) : l ≤ j := by
+  by_contra hn
+  have hz := RankMatching.increment_zero A hA (lt_of_not_ge hn)
+  have ho := ((RankMatching.matched_iff_increments A hA l j).mp h).1
+  omega
+
+def columnMatches (A : Matrix ℤ ℤ k) (J N : ℤ) : Set (ℤ × ℤ) :=
+  {p | RankMatching.Matched A p.1 p.2 ∧ J < p.2 ∧ p.2 ≤ N}
+
+def crossingStageMatches (A : Matrix ℤ ℤ k) (J N : ℤ) : Set (ℤ × ℤ) :=
+  {p | RankMatching.Matched A p.1 p.2 ∧ p.1 ≤ J ∧ J < p.2 ∧ p.2 ≤ N}
+
+lemma columnMatches_snd_inj (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A)
+    (J N : ℤ) : Set.InjOn Prod.snd (columnMatches A J N) := by
+  rintro ⟨l,j⟩ hp ⟨l',j'⟩ hq he
+  change j = j' at he
+  subst j'
+  exact Prod.ext (RankMatching.column_unique A hA hp.1 hq.1) rfl
+
+lemma columnMatches_finite (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A)
+    (J N : ℤ) : (columnMatches A J N).Finite := by
+  exact Set.Finite.of_injOn (fun _ hp => hp.2) (columnMatches_snd_inj A hA J N)
+    (Set.finite_Ioc J N)
+
+lemma columnMatches_ncard (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A)
+    (J N : ℤ) : (columnMatches A J N).ncard =
+      ∑ j ∈ Finset.Ioc J N, if ∃ l, RankMatching.Matched A l j then 1 else 0 := by
+  classical
+  have he : Prod.snd '' columnMatches A J N =
+      ↑((Finset.Ioc J N).filter (fun j => ∃ l, RankMatching.Matched A l j)) := by
+    ext j
+    simp only [Set.mem_image, columnMatches, Set.mem_ofPred_eq, Finset.mem_coe,
+      Finset.mem_filter, Finset.mem_Ioc]
+    constructor
+    · rintro ⟨⟨l,j'⟩,⟨hm,hJ,hN⟩,rfl⟩
+      exact ⟨⟨hJ,hN⟩,l,hm⟩
+    · rintro ⟨⟨hJ,hN⟩,l,hm⟩
+      exact ⟨(l,j),⟨hm,hJ,hN⟩,rfl⟩
+  rw [← (columnMatches_snd_inj A hA J N).ncard_image, he,
+    Set.ncard_coe_finset, Finset.sum_boole]
+  rfl
+
+lemma columnMatches_partition (A : Matrix ℤ ℤ k) (hA : RankMatching.Triangular A)
+    (J N : ℤ) :
+    (columnMatches A J N).ncard = (crossingStageMatches A J N).ncard +
+      RankMatching.blockRank A (J + 1) N := by
+  classical
+  let high := ((Finset.Icc (J + 1) N ×ˢ Finset.Icc (J + 1) N).filter
+    (fun p : ℤ × ℤ => RankMatching.Matched A p.1 p.2))
+  have he : columnMatches A J N = crossingStageMatches A J N ∪ (high : Set (ℤ × ℤ)) := by
+    ext p
+    simp only [columnMatches, crossingStageMatches, Set.mem_ofPred_eq, Set.mem_union,
+      high, Finset.mem_coe, Finset.mem_filter, Finset.mem_product, Finset.mem_Icc]
+    constructor
+    · rintro ⟨hm,hJ,hN⟩
+      by_cases hl : p.1 ≤ J
+      · exact Or.inl ⟨hm,hl,hJ,hN⟩
+      · have hh := matched_le A hA hm
+        exact Or.inr ⟨⟨⟨by omega,by omega⟩,⟨by omega,hN⟩⟩,hm⟩
+    · rintro (⟨hm,hl,hJ,hN⟩ | ⟨⟨hl,hj⟩,hm⟩)
+      · exact ⟨hm,hJ,hN⟩
+      · exact ⟨hm,by omega,hj.2⟩
+  have hd : Disjoint (crossingStageMatches A J N) (high : Set (ℤ × ℤ)) := by
+    apply Set.disjoint_left.mpr
+    intro p hp hq
+    simp only [high, Finset.mem_coe, Finset.mem_filter, Finset.mem_product,
+      Finset.mem_Icc] at hq
+    have hh := hp.2.1
+    omega
+  have hf : (crossingStageMatches A J N).Finite :=
+    (columnMatches_finite A hA J N).subset (fun _ h => ⟨h.1,h.2.2⟩)
+  rw [he, Set.ncard_union_eq hd hf high.finite_toSet, Set.ncard_coe_finset,
+    RankMatching.rank_eq_count A hA]
+  rfl
+
+lemma imageModulo_finrank_partition [Finite k]
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (J N : ℤ) :
+    Module.finrank k (imageModulo a J N) =
+      (crossingStageMatches (matrixOf a) J N).ncard + RankMatching.blockRank (matrixOf a) (J + 1) N := by
+  rw [imageModulo_finrank a ha, ← columnMatches_ncard _ (matrixOf_triangular a ha),
+    columnMatches_partition _ (matrixOf_triangular a ha)]
+
+lemma stageImage_le_ker_projection (a : LaurentSeries k →ₗ[k] LaurentSeries k)
+    (ha : PreservesPoles a) (J : ℤ) : stageImage a J ≤ (poleProjection (J + 1)).ker := by
+  intro f hf
+  change poleProjection (J + 1) f = 0
+  apply poleProjection_eq_zero.mpr
+  simpa using stageImage_le_poleLattice a ha J hf
+
+def quotientProjection (a : LaurentSeries k →ₗ[k] LaurentSeries k)
+    (ha : PreservesPoles a) (J : ℤ) :
+    (LaurentSeries k ⧸ stageImage a J) →ₗ[k] (ℤ → k) :=
+  (stageImage a J).liftQ (poleProjection (J + 1)) (stageImage_le_ker_projection a ha J)
+
+@[simp] lemma quotientProjection_mkQ (a : LaurentSeries k →ₗ[k] LaurentSeries k)
+    (ha : PreservesPoles a) (J : ℤ) (f : LaurentSeries k) :
+    quotientProjection a ha J ((stageImage a J).mkQ f) = poleProjection (J + 1) f := rfl
+
+def crossingStage (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J N : ℤ) :
+    Submodule k (LaurentSeries k ⧸ stageImage a J) :=
+  (stageImage a N ⊓ poleLattice J).map (stageImage a J).mkQ
+
+lemma crossingStage_mono (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
+    Monotone (crossingStage a J) := fun _ _ h =>
+  Submodule.map_mono (inf_le_inf_right _ (stageImage_mono a h))
+
+lemma crossingStage_finite (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J N : ℤ) :
+    FiniteDimensional k (crossingStage a J N) := by
+  let := imageModulo_finite a J N
+  exact Submodule.finiteDimensional_of_le (show crossingStage a J N ≤ imageModulo a J N
+    from Submodule.map_mono inf_le_left)
+
+lemma crossingStage_eq_inf (a : LaurentSeries k →ₗ[k] LaurentSeries k)
+    (ha : PreservesPoles a) (J N : ℤ) :
+    crossingStage a J N = imageModulo a J N ⊓ (quotientProjection a ha J).ker := by
+  ext z
+  constructor
+  · rintro ⟨f,⟨hf,hJ⟩,rfl⟩
+    exact ⟨⟨f,hf,rfl⟩, poleProjection_eq_zero.mpr (by simpa using hJ)⟩
+  · rintro ���⟨f,hf,rfl⟩,hz⟩
+    refine ⟨f,⟨hf,?_⟩,rfl⟩
+    have he : poleProjection (J + 1) f = 0 := hz
+    simpa using poleProjection_eq_zero.mp he
+
+lemma quotientProjection_stage_range (a : LaurentSeries k →ₗ[k] LaurentSeries k)
+    (ha : PreservesPoles a) (J N : ℤ) :
+    ((quotientProjection a ha J).domRestrict (imageModulo a J N)).range =
+      RankMatching.columnSpace (matrixOf a) (J + 1) N := by
+  rw [LinearMap.range_domRestrict, columnSpace_eq_projectedImage a ha]
+  ext v
+  constructor
+  · rintro ⟨z,⟨f,hf,rfl⟩,rfl⟩
+    exact ⟨f,hf,rfl⟩
+  · rintro ⟨f,hf,rfl⟩
+    exact ⟨(stageImage a J).mkQ f, ⟨f,hf,rfl⟩,rfl⟩
+
+lemma crossingStage_finrank [Finite k]
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a) (J N : ℤ) :
+    Module.finrank k (crossingStage a J N) = (crossingStageMatches (matrixOf a) J N).ncard := by
+  let := imageModulo_finite a J N
+  let f := (quotientProjection a ha J).domRestrict (imageModulo a J N)
+  have hr := LinearMap.finrank_range_add_finrank_ker f
+  have hk : Module.finrank k f.ker = Module.finrank k (crossingStage a J N) := by
+    rw [← Submodule.finrank_map_subtype_eq (imageModulo a J N) f.ker,
+      show f.ker = ((quotientProjection a ha J).ker).comap (imageModulo a J N).subtype
+        from LinearMap.ker_domRestrict _ _,
+      Submodule.map_comap_subtype, ← crossingStage_eq_inf a ha]
+  have hi : Module.finrank k f.range = RankMatching.blockRank (matrixOf a) (J + 1) N := by
+    rw [quotientProjection_stage_range, ← RankMatching.blockRank_eq_finrank _ (matrixOf_triangular a ha)]
+  rw [hk,hi,imageModulo_finrank_partition a ha] at hr
+  omega
+
+def crossingSpace (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
+    Submodule k (LaurentSeries k ⧸ stageImage a J) :=
+  (a.range ⊓ poleLattice J).map (stageImage a J).mkQ
+
+lemma crossingStage_le_space (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J N : ℤ) :
+    crossingStage a J N ≤ crossingSpace a J := by
+  apply Submodule.map_mono
+  apply inf_le_inf_right
+  rintro f ⟨g,hg,rfl⟩
+  exact ⟨g,rfl⟩
+
+lemma crossingSpace_stabilizes [Finite k]
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a)
+    (J : ℤ) (hc : (crossing a J).Finite) :
+    ∃ N : ℤ, J ≤ N ∧ crossingSpace a J = crossingStage a J N ∧
+      crossingStageMatches (matrixOf a) J N = crossing a J := by
+  obtain ⟨b,hb⟩ := (hc.image Prod.snd).bddAbove
+  let N := max J b
+  have he (M : ℤ) (hM : N ≤ M) :
+      crossingStageMatches (matrixOf a) J M = crossing a J := by
+    ext p
+    constructor
+    · intro hp
+      exact ⟨hp.1,hp.2.1,hp.2.2.1⟩
+    · intro hp
+      exact ⟨hp.1,hp.2.1,hp.2.2,le_trans (hb ⟨p,hp,rfl⟩) (le_trans (le_max_right J b) hM)⟩
+  refine ⟨N, le_max_left _ _, ?_, he N le_rfl⟩
+  apply le_antisymm _ (crossingStage_le_space a J N)
+  rintro z ⟨f,⟨⟨x,hx⟩,hf⟩,rfl⟩
+  obtain ⟨M,hM⟩ := exists_poleLattice x
+  have heq : crossingStage a J N = crossingStage a J (max N M) := by
+    let := crossingStage_finite a J (max N M)
+    apply Submodule.eq_of_le_of_finrank_eq (crossingStage_mono a J (le_max_left _ _))
+    rw [crossingStage_finrank a ha, crossingStage_finrank a ha,
+      he N le_rfl, he (max N M) (le_max_left _ _)]
+  rw [heq]
+  exact ⟨f,⟨⟨x,poleLattice_mono (le_max_right N M) hM,hx⟩,hf⟩,rfl⟩
+
+def crossingQuotientEquiv (a : LaurentSeries k →ₗ[k] LaurentSeries k) (J : ℤ) :
+    crossingQuotient a J ≃ₗ[k] crossingSpace a J := by
+  let I := a.range ⊓ poleLattice (k := k) J
+  let f := (stageImage a J).mkQ.domRestrict I
+  have hk : f.ker = (stageImage a J).comap I.subtype := by
+    rw [LinearMap.ker_domRestrict, Submodule.ker_mkQ]
+  exact (Submodule.quotEquivOfEq _ _ hk.symm).trans
+    (f.quotKerEquivRange.trans (LinearEquiv.ofEq _ _ (LinearMap.range_domRestrict _ _)))
+
+theorem crossing_dimension [Finite k]
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a)
+    (hc : ∀ J, (crossing a J).Finite) (J : ℤ) :
+    FiniteDimensional k (crossingQuotient a J) ∧
+      Module.finrank k (crossingQuotient a J) = (crossing a J).ncard := by
+  obtain ⟨N,hN,he,hcN⟩ := crossingSpace_stabilizes a ha J (hc J)
+  have : FiniteDimensional k (crossingSpace a J) := by
+    rw [he]
+    exact crossingStage_finite a J N
+  refine ⟨(crossingQuotientEquiv a J).symm.finiteDimensional, ?_⟩
+  rw [(crossingQuotientEquiv a J).finrank_eq, he, crossingStage_finrank a ha, hcN]
+
+lemma range_inf_stabilizes [Finite k]
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a)
+    (J : ℤ) (hc : (crossing a J).Finite) :
+    ∃ N : ℤ, a.range ⊓ poleLattice J = stageImage a N ⊓ poleLattice J := by
+  obtain ⟨N,hN,he,_⟩ := crossingSpace_stabilizes a ha J hc
+  refine ⟨N,le_antisymm ?_ ?_⟩
+  · intro f hf
+    have hq : (stageImage a J).mkQ f ∈ crossingStage a J N := by
+      rw [← he]
+      exact ⟨f,hf,rfl⟩
+    have hi : (stageImage a J).mkQ f ∈ imageModulo a J N :=
+      (Submodule.map_mono (show stageImage a N ⊓ poleLattice J ≤ stageImage a N
+        from inf_le_left)) hq
+    exact ⟨(mem_imageModulo a hN f).mp hi,hf.2⟩
+  · rintro f ⟨⟨x,hx,rfl⟩,hf⟩
+    exact ⟨⟨x,rfl⟩,hf⟩
+
+lemma isCompact_range_inter_poleLattice [Finite k]
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a)
+    (J : ℤ) (hc : (crossing a J).Finite) :
+    IsCompact ((Set.range a) ∩ (poleLattice (k := k) J : Set (LaurentSeries k))) := by
+  obtain ⟨N,he⟩ := range_inf_stabilizes a ha J hc
+  change IsCompact (↑(a.range ⊓ poleLattice J) : Set (LaurentSeries k))
+  rw [he]
+  change IsCompact ((a '' (poleLattice (k := k) N : Set (LaurentSeries k))) ∩
+    (poleLattice (k := k) J : Set (LaurentSeries k)))
+  exact ((isCompact_poleLattice N).image (continuous_of_preservesPoles a ha)).inter_right
+    (isClosed_poleLattice J)
+
+theorem closed_image [Finite k]
+    (a : LaurentSeries k →ₗ[k] LaurentSeries k) (ha : PreservesPoles a)
+    (hc : ∀ J, (crossing a J).Finite) :
+    IsClosed (Set.range a) := by
+  rw [← isOpen_compl_iff, isOpen_iff_forall_mem_open]
+  intro f hf
+  obtain ⟨J,hJ⟩ := exists_poleLattice f
+  refine ⟨(poleLattice (k := k) J : Set (LaurentSeries k)) ∩
+    ((Set.range a) ∩ (poleLattice (k := k) J : Set (LaurentSeries k)))ᶜ, ?_, ?_, ?_⟩
+  · intro g hg hga
+    exact hg.2 ⟨hga,hg.1⟩
+  · exact (isOpen_poleLattice J).inter
+      (isCompact_range_inter_poleLattice a ha J (hc J)).isClosed.isOpen_compl
+  · exact ⟨hJ,fun h => hf h.1⟩
+
+end HahnWilson.PivotTopology
+
+end
+
+end OAI

lean/OAI/Topology/Homotopy/BasedMaps.lean

@@ -1,71 +1,71 @@
-import Mathlib
-
-namespace OAI
-
-/-! Maps of based generalized loops and homotopy groups. -/
-
-noncomputable section
-universe u v w
-
-
-namespace NearbyLagrangian.BasedHomotopy
-open scoped Topology Topology.Homotopy
-variable {N X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
-variable {x : X} {y : Y} {z : Z}
-
-def loopMap (e : C(X,Y)) (he : e x = y) (f : GenLoop N X x) : GenLoop N Y y :=
-  ⟨e.comp f.1,fun t ht => (congrArg e (GenLoop.boundary f t ht)).trans he⟩
-
-lemma loopMap_homotopic (e : C(X,Y)) (he : e x = y) {f g : GenLoop N X x}
-    (h : GenLoop.Homotopic f g) : GenLoop.Homotopic (loopMap e he f) (loopMap e he g) := by
-  obtain ⟨H⟩ := h
-  exact ⟨H.compContinuousMap e⟩
-
-@[simp] theorem loopMap_const (e : C(X,Y)) (he : e x = y) :
-    loopMap (N:=N) e he GenLoop.const = GenLoop.const := by ext t; exact he
-
-@[simp] theorem loopMap_id (f : GenLoop N X x) : loopMap (ContinuousMap.id X) rfl f = f := by
-  ext t; rfl
-
-@[simp] theorem loopMap_comp (e : C(X,Y)) (he : e x = y) (d : C(Y,Z)) (hd : d y = z)
-    (f : GenLoop N X x) : loopMap d hd (loopMap e he f) =
-      loopMap (d.comp e) ((congrArg d he).trans hd) f := by
-  ext t; rfl
-
-lemma loopMap_transAt [DecidableEq N] (e : C(X,Y)) (he : e x = y)
-    (i : N) (f g : GenLoop N X x) : loopMap e he (GenLoop.transAt i f g) =
-      GenLoop.transAt i (loopMap e he f) (loopMap e he g) := by
-  ext t
-  simp only [loopMap,GenLoop.mk_apply,ContinuousMap.comp_apply,GenLoop.transAt,GenLoop.coe_copy,GenLoop.coe_coe]
-  split_ifs <;> rfl
-
-def groupMap [DecidableEq N] [Nonempty N] (e : C(X,Y)) (he : e x = y) :
-    HomotopyGroup N X x →* HomotopyGroup N Y y where
-  toFun := Quotient.map (loopMap e he) (fun _ _ h => loopMap_homotopic e he h)
-  map_one' := by
-    rw [HomotopyGroup.one_def,HomotopyGroup.one_def]
-    change Quotient.mk'' (loopMap e he GenLoop.const) = Quotient.mk'' GenLoop.const
-    rw [loopMap_const]
-  map_mul' a b := Quotient.inductionOn₂ a b fun f g => by
-    simp only [HomotopyGroup.mul_spec (i:=Classical.arbitrary N)]
-    change (⟦loopMap e he (GenLoop.transAt (Classical.arbitrary N) g f)⟧ : HomotopyGroup N Y y) =
-      ((· * ·) : HomotopyGroup N Y y → HomotopyGroup N Y y → HomotopyGroup N Y y)
-        ⟦loopMap e he f⟧ ⟦loopMap e he g⟧
-    rw [HomotopyGroup.mul_spec (i:=Classical.arbitrary N),loopMap_transAt]
-
-@[simp] theorem groupMap_id [DecidableEq N] [Nonempty N] (a : HomotopyGroup N X x) :
-    groupMap (ContinuousMap.id X) rfl a = a := by
-  induction a using Quotient.inductionOn with
-  | h f => change Quotient.mk'' (loopMap _ _ f) = _; rw [loopMap_id]; rfl
-
-@[simp] theorem groupMap_comp [DecidableEq N] [Nonempty N]
-    (e : C(X,Y)) (he : e x = y) (d : C(Y,Z)) (hd : d y = z) (a : HomotopyGroup N X x) :
-    groupMap d hd (groupMap e he a) = groupMap (d.comp e) ((congrArg d he).trans hd) a := by
-  induction a using Quotient.inductionOn with
-  | h f => change Quotient.mk'' (loopMap _ _ (loopMap _ _ f)) = _; rw [loopMap_comp]; rfl
-
-end NearbyLagrangian.BasedHomotopy
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Maps of based generalized loops and homotopy groups. -/
+
+noncomputable section
+universe u v w
+
+
+namespace NearbyLagrangian.BasedHomotopy
+open scoped Topology Topology.Homotopy
+variable {N X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z]
+variable {x : X} {y : Y} {z : Z}
+
+def loopMap (e : C(X,Y)) (he : e x = y) (f : GenLoop N X x) : GenLoop N Y y :=
+  ⟨e.comp f.1,fun t ht => (congrArg e (GenLoop.boundary f t ht)).trans he⟩
+
+lemma loopMap_homotopic (e : C(X,Y)) (he : e x = y) {f g : GenLoop N X x}
+    (h : GenLoop.Homotopic f g) : GenLoop.Homotopic (loopMap e he f) (loopMap e he g) := by
+  obtain ⟨H⟩ := h
+  exact ⟨H.compContinuousMap e⟩
+
+@[simp] theorem loopMap_const (e : C(X,Y)) (he : e x = y) :
+    loopMap (N:=N) e he GenLoop.const = GenLoop.const := by ext t; exact he
+
+@[simp] theorem loopMap_id (f : GenLoop N X x) : loopMap (ContinuousMap.id X) rfl f = f := by
+  ext t; rfl
+
+@[simp] theorem loopMap_comp (e : C(X,Y)) (he : e x = y) (d : C(Y,Z)) (hd : d y = z)
+    (f : GenLoop N X x) : loopMap d hd (loopMap e he f) =
+      loopMap (d.comp e) ((congrArg d he).trans hd) f := by
+  ext t; rfl
+
+lemma loopMap_transAt [DecidableEq N] (e : C(X,Y)) (he : e x = y)
+    (i : N) (f g : GenLoop N X x) : loopMap e he (GenLoop.transAt i f g) =
+      GenLoop.transAt i (loopMap e he f) (loopMap e he g) := by
+  ext t
+  simp only [loopMap,GenLoop.mk_apply,ContinuousMap.comp_apply,GenLoop.transAt,GenLoop.coe_copy,GenLoop.coe_coe]
+  split_ifs <;> rfl
+
+def groupMap [DecidableEq N] [Nonempty N] (e : C(X,Y)) (he : e x = y) :
+    HomotopyGroup N X x →* HomotopyGroup N Y y where
+  toFun := Quotient.map (loopMap e he) (fun _ _ h => loopMap_homotopic e he h)
+  map_one' := by
+    rw [HomotopyGroup.one_def,HomotopyGroup.one_def]
+    change Quotient.mk'' (loopMap e he GenLoop.const) = Quotient.mk'' GenLoop.const
+    rw [loopMap_const]
+  map_mul' a b := Quotient.inductionOn₂ a b fun f g => by
+    simp only [HomotopyGroup.mul_spec (i:=Classical.arbitrary N)]
+    change (⟦loopMap e he (GenLoop.transAt (Classical.arbitrary N) g f)⟧ : HomotopyGroup N Y y) =
+      ((· * ·) : HomotopyGroup N Y y → HomotopyGroup N Y y → HomotopyGroup N Y y)
+        ⟦loopMap e he f⟧ ⟦loopMap e he g⟧
+    rw [HomotopyGroup.mul_spec (i:=Classical.arbitrary N),loopMap_transAt]
+
+@[simp] theorem groupMap_id [DecidableEq N] [Nonempty N] (a : HomotopyGroup N X x) :
+    groupMap (ContinuousMap.id X) rfl a = a := by
+  induction a using Quotient.inductionOn with
+  | h f => change Quotient.mk'' (loopMap _ _ f) = _; rw [loopMap_id]; rfl
+
+@[simp] theorem groupMap_comp [DecidableEq N] [Nonempty N]
+    (e : C(X,Y)) (he : e x = y) (d : C(Y,Z)) (hd : d y = z) (a : HomotopyGroup N X x) :
+    groupMap d hd (groupMap e he a) = groupMap (d.comp e) ((congrArg d he).trans hd) a := by
+  induction a using Quotient.inductionOn with
+  | h f => change Quotient.mk'' (loopMap _ _ (loopMap _ _ f)) = _; rw [loopMap_comp]; rfl
+
+end NearbyLagrangian.BasedHomotopy
+
+end
+
+end OAI

lean/OAI/Topology/Homotopy/ConeHomotopy.lean

@@ -1,234 +1,234 @@
-import Mathlib
-import OAI.Topology.Homotopy.FiberCone
-
-namespace OAI
-
-/-! Right suspension homotopies and suspension of maps and homotopies. -/
-
-noncomputable section
-universe u v w
-open Set ContinuousMap
-open scoped Topology unitInterval
-
-namespace NearbyLagrangian.FiberCone
-variable {B A X : Type*} [TopologicalSpace B] [TopologicalSpace A] [TopologicalSpace X]
-variable (f : C(A,X)) (p : C(X,B)) (q : C(A,B)) (hf : ∀ a, p (f a) = q a)
-
-def sweepBase : C(unitInterval × X,Susp p) := meridian p
-
-def sweepRay : C(unitInterval × (unitInterval × A),Susp p) where
-  toFun z := meridian p (max z.2.1 z.1,f z.2.2)
-  continuous_toFun := (meridian p).continuous.comp
-    (((continuous_fst.comp continuous_snd).max continuous_fst).prodMk
-      (f.continuous.comp (continuous_snd.comp continuous_snd)))
-
-def sweepApex : C(unitInterval × B,Susp p) := (north p).comp ⟨Prod.snd,continuous_snd⟩
-
-lemma sweepRay_bottom (s : unitInterval) (a : A) :
-    sweepRay f p (s,(0,a)) = sweepBase p (s,f a) := by
-  change meridian p (max 0 s,f a) = meridian p (s,f a)
-  rw [max_eq_right (show (0:unitInterval) ≤ s from bot_le)]
-
-include hf in
-lemma sweepRay_top (s : unitInterval) (a : A) :
-    sweepRay f p (s,(1,a)) = sweepApex p (s,q a) := by
-  change meridian p (max 1 s,f a) = north p (q a)
-  rw [max_eq_left (show s ≤ (1:unitInterval) from le_top),ray_one,hf]
-
-def sweep : C(unitInterval × Space f q,Susp p) :=
-  cylinderLift f q (sweepBase p) (sweepRay f p) (sweepApex p)
-    (sweepRay_bottom f p) (sweepRay_top f p q hf)
-
-@[simp] lemma sweep_base (s : unitInterval) (x : X) :
-    sweep f p q hf (s,base f q x) = meridian p (s,x) := rfl
-@[simp] lemma sweep_ray (s : unitInterval) (ta : unitInterval × A) :
-    sweep f p q hf (s,ray f q ta) = meridian p (max ta.1 s,f ta.2) := rfl
-@[simp] lemma sweep_apex (s : unitInterval) (b : B) :
-    sweep f p q hf (s,apex f q b) = north p b := rfl
-
-lemma sweep_zero (z : Space f q) :
-    sweep f p q hf (0,z) = suspMap f p q hf (collapse f p q hf z) := by
-  induction z using induction with
-  | hbase x => rw [sweep_base,ray_zero,collapse_base,suspMap_south]
-  | hray ta =>
-    rw [sweep_ray,max_eq_left (show (0:unitInterval) ≤ ta.1 from bot_le),
-      collapse_ray,suspMap_meridian]
-  | hapex b => rw [sweep_apex,collapse_apex,suspMap_north]
-
-lemma sweep_one (z : Space f q) :
-    sweep f p q hf (1,z) = north p (projection f p q hf z) := by
-  induction z using induction with
-  | hbase x => rw [sweep_base,ray_one,projection_base]
-  | hray ta =>
-    rw [sweep_ray,max_eq_right (show ta.1 ≤ (1:unitInterval) from le_top),
-      ray_one,hf,projection_ray]
-  | hapex b => rfl
-
-lemma sweep_over (s : unitInterval) (z : Space f q) :
-    suspProjection p (sweep f p q hf (s,z)) = projection f p q hf z := by
-  induction z using induction with
-  | hbase x => rfl
-  | hray ta => exact hf ta.2
-  | hapex b => rfl
-
-namespace Contraction
-variable {f p q hf} (H : Contraction f p q hf)
-
-def rightRay : C(unitInterval × (unitInterval × X),Susp p) where
-  toFun z := sweep f p q hf (horizontal (z.1,z.2.1),
-    H.hom (vertical (z.1,z.2.1),base f q z.2.2))
-  continuous_toFun := (sweep f p q hf).continuous.comp
-    ((horizontal.continuous.comp (continuous_fst.prodMk (continuous_fst.comp continuous_snd))).prodMk
-      (H.hom.continuous.comp
-        ((vertical.continuous.comp (continuous_fst.prodMk (continuous_fst.comp continuous_snd))).prodMk
-          ((base f q).continuous.comp (continuous_snd.comp continuous_snd)))))
-
-lemma rightRay_bottom (s : unitInterval) (x : X) : H.rightRay (s,(0,x)) = south p (p x) := by
-  change sweep f p q hf (horizontal (s,0),H.hom (vertical (s,0),base f q x)) = _
-  rw [horizontal_bottom,vertical_bottom,H.at_zero,sweep_base,ray_zero]
-
-lemma rightRay_top (s : unitInterval) (x : X) : H.rightRay (s,(1,x)) = north p (p x) := by
-  change sweep f p q hf (horizontal (s,1),H.hom (vertical (s,1),base f q x)) = _
-  rcases square_top s with hx | hy
-  · rw [hx,sweep_one,H.staysOver,projection_base]
-  · rw [hy,H.at_one,projection_base,sweep_apex]
-
-def rightDeformation : C(unitInterval × Susp p,Susp p) :=
-  cylinderLift p p ((south p).comp ⟨Prod.snd,continuous_snd⟩) H.rightRay
-    ((north p).comp ⟨Prod.snd,continuous_snd⟩) H.rightRay_bottom H.rightRay_top
-
-@[simp] lemma rightDeformation_south (s : unitInterval) (b : B) :
-    H.rightDeformation (s,south p b) = south p b := rfl
-@[simp] lemma rightDeformation_meridian (s : unitInterval) (tx : unitInterval × X) :
-    H.rightDeformation (s,meridian p tx) = H.rightRay (s,tx) := rfl
-@[simp] lemma rightDeformation_north (s : unitInterval) (b : B) :
-    H.rightDeformation (s,north p b) = north p b := rfl
-
-lemma rightDeformation_zero (z : Susp p) : H.rightDeformation (0,z) = z := by
-  induction z using induction with
-  | hbase b => rfl
-  | hray tx =>
-    rw [rightDeformation_meridian]
-    change sweep f p q hf (horizontal (0,tx.1),
-      H.hom (vertical (0,tx.1),base f q tx.2)) = _
-    rw [horizontal_left,vertical_left,H.at_zero,sweep_base]
-  | hapex b => rfl
-
-lemma rightDeformation_one (z : Susp p) :
-    H.rightDeformation (1,z) = suspMap f p q hf (H.inverse z) := by
-  induction z using induction with
-  | hbase b => rw [rightDeformation_south,inverse_south,suspMap_south]
-  | hray tx =>
-    rw [rightDeformation_meridian,inverse_meridian]
-    change sweep f p q hf (horizontal (1,tx.1),
-      H.hom (vertical (1,tx.1),base f q tx.2)) = _
-    rw [horizontal_right,vertical_right,sweep_zero]
-  | hapex b => rw [rightDeformation_north,inverse_north,suspMap_north]
-
-lemma rightDeformation_over (s : unitInterval) (z : Susp p) :
-    suspProjection p (H.rightDeformation (s,z)) = suspProjection p z := by
-  induction z using induction with
-  | hbase b => rfl
-  | hray tx =>
-    rw [rightDeformation_meridian]
-    change suspProjection p (sweep f p q hf (horizontal (s,tx.1),
-      H.hom (vertical (s,tx.1),base f q tx.2))) = _
-    rw [sweep_over,H.staysOver,projection_base]
-    rfl
-  | hapex b => rfl
-
-def rightHomotopy : ((suspMap f p q hf).comp H.inverse).Homotopy (ContinuousMap.id (Susp p)) :=
-  ContinuousMap.Homotopy.symm
-    { toContinuousMap := H.rightDeformation
-      map_zero_left := H.rightDeformation_zero
-      map_one_left := H.rightDeformation_one }
-
-lemma rightHomotopy_over (s : unitInterval) (z : Susp p) :
-    suspProjection p (H.rightHomotopy (s,z)) = suspProjection p z :=
-  H.rightDeformation_over (unitInterval.symm s) z
-
-end Contraction
-end NearbyLagrangian.FiberCone
-
-end
-
-noncomputable section
-universe u v w
-open Set ContinuousMap
-open scoped Topology unitInterval
-
-namespace NearbyLagrangian.FiberCone
-variable {B A X Y : Type*} [TopologicalSpace B] [TopologicalSpace A]
-  [TopologicalSpace X] [TopologicalSpace Y]
-
-@[simp] lemma suspMap_id (q : C(A,B)) :
-    suspMap (ContinuousMap.id A) q q (fun _ => rfl) = ContinuousMap.id (Susp q) := by
-  ext z
-  induction z using induction with
-  | hbase _ => rfl
-  | hray _ => rfl
-  | hapex _ => rfl
-
-lemma suspMap_comp (f : C(A,X)) (g : C(X,Y)) (p : C(X,B)) (q : C(A,B)) (r : C(Y,B))
-    (hf : ∀ a, p (f a) = q a) (hg : ∀ x, r (g x) = p x) :
-    (suspMap g r p hg).comp (suspMap f p q hf) =
-      suspMap (g.comp f) r q (fun a => (hg (f a)).trans (hf a)) := by
-  ext z
-  induction z using induction with
-  | hbase _ => rfl
-  | hray _ => rfl
-  | hapex _ => rfl
-
-variable {p : C(X,B)} {q : C(A,B)} {f g : C(A,X)}
-  (hf : ∀ a, p (f a) = q a) (hg : ∀ a, p (g a) = q a)
-  (H : f.Homotopy g) (hH : ∀ s a, p (H (s,a)) = q a)
-
-def suspHomotopyRay : C(unitInterval × (unitInterval × A),Susp p) :=
-  (meridian p).comp ⟨fun z => (z.2.1,H (z.1,z.2.2)),
-    (continuous_fst.comp continuous_snd).prodMk
-      (H.continuous.comp (continuous_fst.prodMk (continuous_snd.comp continuous_snd)))⟩
-
-include hH in
-lemma suspHomotopyRay_bottom (s : unitInterval) (a : A) :
-    suspHomotopyRay H (s,(0,a)) = south p (q a) := by
-  change meridian p (0,H (s,a)) = _
-  rw [ray_zero,hH]
-
-include hH in
-lemma suspHomotopyRay_top (s : unitInterval) (a : A) :
-    suspHomotopyRay H (s,(1,a)) = north p (q a) := by
-  change meridian p (1,H (s,a)) = _
-  rw [ray_one,hH]
-
-def suspHomotopy : (suspMap f p q hf).Homotopy (suspMap g p q hg) where
-  toContinuousMap := cylinderLift q q
-    ((south p).comp ⟨Prod.snd,continuous_snd⟩) (suspHomotopyRay H)
-    ((north p).comp ⟨Prod.snd,continuous_snd⟩)
-    (suspHomotopyRay_bottom H hH) (suspHomotopyRay_top H hH)
-  map_zero_left z := by
-    induction z using induction with
-    | hbase _ => rfl
-    | hray ta =>
-      change meridian p (ta.1,H (0,ta.2)) = meridian p (ta.1,f ta.2)
-      rw [HomotopyLike.map_zero_left H]
-    | hapex _ => rfl
-  map_one_left z := by
-    induction z using induction with
-    | hbase _ => rfl
-    | hray ta =>
-      change meridian p (ta.1,H (1,ta.2)) = meridian p (ta.1,g ta.2)
-      rw [HomotopyLike.map_one_left H]
-    | hapex _ => rfl
-
-lemma suspHomotopy_over (s : unitInterval) (z : Susp q) :
-    suspProjection p (suspHomotopy hf hg H hH (s,z)) = suspProjection q z := by
-  induction z using induction with
-  | hbase _ => rfl
-  | hray ta => exact hH s ta.2
-  | hapex _ => rfl
-
-end NearbyLagrangian.FiberCone
-
-end
-
-end OAI
+import Mathlib
+import OAI.Topology.Homotopy.FiberCone
+
+namespace OAI
+
+/-! Right suspension homotopies and suspension of maps and homotopies. -/
+
+noncomputable section
+universe u v w
+open Set ContinuousMap
+open scoped Topology unitInterval
+
+namespace NearbyLagrangian.FiberCone
+variable {B A X : Type*} [TopologicalSpace B] [TopologicalSpace A] [TopologicalSpace X]
+variable (f : C(A,X)) (p : C(X,B)) (q : C(A,B)) (hf : ∀ a, p (f a) = q a)
+
+def sweepBase : C(unitInterval × X,Susp p) := meridian p
+
+def sweepRay : C(unitInterval × (unitInterval × A),Susp p) where
+  toFun z := meridian p (max z.2.1 z.1,f z.2.2)
+  continuous_toFun := (meridian p).continuous.comp
+    (((continuous_fst.comp continuous_snd).max continuous_fst).prodMk
+      (f.continuous.comp (continuous_snd.comp continuous_snd)))
+
+def sweepApex : C(unitInterval × B,Susp p) := (north p).comp ⟨Prod.snd,continuous_snd⟩
+
+lemma sweepRay_bottom (s : unitInterval) (a : A) :
+    sweepRay f p (s,(0,a)) = sweepBase p (s,f a) := by
+  change meridian p (max 0 s,f a) = meridian p (s,f a)
+  rw [max_eq_right (show (0:unitInterval) ≤ s from bot_le)]
+
+include hf in
+lemma sweepRay_top (s : unitInterval) (a : A) :
+    sweepRay f p (s,(1,a)) = sweepApex p (s,q a) := by
+  change meridian p (max 1 s,f a) = north p (q a)
+  rw [max_eq_left (show s ≤ (1:unitInterval) from le_top),ray_one,hf]
+
+def sweep : C(unitInterval × Space f q,Susp p) :=
+  cylinderLift f q (sweepBase p) (sweepRay f p) (sweepApex p)
+    (sweepRay_bottom f p) (sweepRay_top f p q hf)
+
+@[simp] lemma sweep_base (s : unitInterval) (x : X) :
+    sweep f p q hf (s,base f q x) = meridian p (s,x) := rfl
+@[simp] lemma sweep_ray (s : unitInterval) (ta : unitInterval × A) :
+    sweep f p q hf (s,ray f q ta) = meridian p (max ta.1 s,f ta.2) := rfl
+@[simp] lemma sweep_apex (s : unitInterval) (b : B) :
+    sweep f p q hf (s,apex f q b) = north p b := rfl
+
+lemma sweep_zero (z : Space f q) :
+    sweep f p q hf (0,z) = suspMap f p q hf (collapse f p q hf z) := by
+  induction z using induction with
+  | hbase x => rw [sweep_base,ray_zero,collapse_base,suspMap_south]
+  | hray ta =>
+    rw [sweep_ray,max_eq_left (show (0:unitInterval) ≤ ta.1 from bot_le),
+      collapse_ray,suspMap_meridian]
+  | hapex b => rw [sweep_apex,collapse_apex,suspMap_north]
+
+lemma sweep_one (z : Space f q) :
+    sweep f p q hf (1,z) = north p (projection f p q hf z) := by
+  induction z using induction with
+  | hbase x => rw [sweep_base,ray_one,projection_base]
+  | hray ta =>
+    rw [sweep_ray,max_eq_right (show ta.1 ≤ (1:unitInterval) from le_top),
+      ray_one,hf,projection_ray]
+  | hapex b => rfl
+
+lemma sweep_over (s : unitInterval) (z : Space f q) :
+    suspProjection p (sweep f p q hf (s,z)) = projection f p q hf z := by
+  induction z using induction with
+  | hbase x => rfl
+  | hray ta => exact hf ta.2
+  | hapex b => rfl
+
+namespace Contraction
+variable {f p q hf} (H : Contraction f p q hf)
+
+def rightRay : C(unitInterval × (unitInterval × X),Susp p) where
+  toFun z := sweep f p q hf (horizontal (z.1,z.2.1),
+    H.hom (vertical (z.1,z.2.1),base f q z.2.2))
+  continuous_toFun := (sweep f p q hf).continuous.comp
+    ((horizontal.continuous.comp (continuous_fst.prodMk (continuous_fst.comp continuous_snd))).prodMk
+      (H.hom.continuous.comp
+        ((vertical.continuous.comp (continuous_fst.prodMk (continuous_fst.comp continuous_snd))).prodMk
+          ((base f q).continuous.comp (continuous_snd.comp continuous_snd)))))
+
+lemma rightRay_bottom (s : unitInterval) (x : X) : H.rightRay (s,(0,x)) = south p (p x) := by
+  change sweep f p q hf (horizontal (s,0),H.hom (vertical (s,0),base f q x)) = _
+  rw [horizontal_bottom,vertical_bottom,H.at_zero,sweep_base,ray_zero]
+
+lemma rightRay_top (s : unitInterval) (x : X) : H.rightRay (s,(1,x)) = north p (p x) := by
+  change sweep f p q hf (horizontal (s,1),H.hom (vertical (s,1),base f q x)) = _
+  rcases square_top s with hx | hy
+  · rw [hx,sweep_one,H.staysOver,projection_base]
+  · rw [hy,H.at_one,projection_base,sweep_apex]
+
+def rightDeformation : C(unitInterval × Susp p,Susp p) :=
+  cylinderLift p p ((south p).comp ⟨Prod.snd,continuous_snd⟩) H.rightRay
+    ((north p).comp ⟨Prod.snd,continuous_snd⟩) H.rightRay_bottom H.rightRay_top
+
+@[simp] lemma rightDeformation_south (s : unitInterval) (b : B) :
+    H.rightDeformation (s,south p b) = south p b := rfl
+@[simp] lemma rightDeformation_meridian (s : unitInterval) (tx : unitInterval × X) :
+    H.rightDeformation (s,meridian p tx) = H.rightRay (s,tx) := rfl
+@[simp] lemma rightDeformation_north (s : unitInterval) (b : B) :
+    H.rightDeformation (s,north p b) = north p b := rfl
+
+lemma rightDeformation_zero (z : Susp p) : H.rightDeformation (0,z) = z := by
+  induction z using induction with
+  | hbase b => rfl
+  | hray tx =>
+    rw [rightDeformation_meridian]
+    change sweep f p q hf (horizontal (0,tx.1),
+      H.hom (vertical (0,tx.1),base f q tx.2)) = _
+    rw [horizontal_left,vertical_left,H.at_zero,sweep_base]
+  | hapex b => rfl
+
+lemma rightDeformation_one (z : Susp p) :
+    H.rightDeformation (1,z) = suspMap f p q hf (H.inverse z) := by
+  induction z using induction with
+  | hbase b => rw [rightDeformation_south,inverse_south,suspMap_south]
+  | hray tx =>
+    rw [rightDeformation_meridian,inverse_meridian]
+    change sweep f p q hf (horizontal (1,tx.1),
+      H.hom (vertical (1,tx.1),base f q tx.2)) = _
+    rw [horizontal_right,vertical_right,sweep_zero]
+  | hapex b => rw [rightDeformation_north,inverse_north,suspMap_north]
+
+lemma rightDeformation_over (s : unitInterval) (z : Susp p) :
+    suspProjection p (H.rightDeformation (s,z)) = suspProjection p z := by
+  induction z using induction with
+  | hbase b => rfl
+  | hray tx =>
+    rw [rightDeformation_meridian]
+    change suspProjection p (sweep f p q hf (horizontal (s,tx.1),
+      H.hom (vertical (s,tx.1),base f q tx.2))) = _
+    rw [sweep_over,H.staysOver,projection_base]
+    rfl
+  | hapex b => rfl
+
+def rightHomotopy : ((suspMap f p q hf).comp H.inverse).Homotopy (ContinuousMap.id (Susp p)) :=
+  ContinuousMap.Homotopy.symm
+    { toContinuousMap := H.rightDeformation
+      map_zero_left := H.rightDeformation_zero
+      map_one_left := H.rightDeformation_one }
+
+lemma rightHomotopy_over (s : unitInterval) (z : Susp p) :
+    suspProjection p (H.rightHomotopy (s,z)) = suspProjection p z :=
+  H.rightDeformation_over (unitInterval.symm s) z
+
+end Contraction
+end NearbyLagrangian.FiberCone
+
+end
+
+noncomputable section
+universe u v w
+open Set ContinuousMap
+open scoped Topology unitInterval
+
+namespace NearbyLagrangian.FiberCone
+variable {B A X Y : Type*} [TopologicalSpace B] [TopologicalSpace A]
+  [TopologicalSpace X] [TopologicalSpace Y]
+
+@[simp] lemma suspMap_id (q : C(A,B)) :
+    suspMap (ContinuousMap.id A) q q (fun _ => rfl) = ContinuousMap.id (Susp q) := by
+  ext z
+  induction z using induction with
+  | hbase _ => rfl
+  | hray _ => rfl
+  | hapex _ => rfl
+
+lemma suspMap_comp (f : C(A,X)) (g : C(X,Y)) (p : C(X,B)) (q : C(A,B)) (r : C(Y,B))
+    (hf : ∀ a, p (f a) = q a) (hg : ∀ x, r (g x) = p x) :
+    (suspMap g r p hg).comp (suspMap f p q hf) =
+      suspMap (g.comp f) r q (fun a => (hg (f a)).trans (hf a)) := by
+  ext z
+  induction z using induction with
+  | hbase _ => rfl
+  | hray _ => rfl
+  | hapex _ => rfl
+
+variable {p : C(X,B)} {q : C(A,B)} {f g : C(A,X)}
+  (hf : ∀ a, p (f a) = q a) (hg : ∀ a, p (g a) = q a)
+  (H : f.Homotopy g) (hH : ∀ s a, p (H (s,a)) = q a)
+
+def suspHomotopyRay : C(unitInterval × (unitInterval × A),Susp p) :=
+  (meridian p).comp ⟨fun z => (z.2.1,H (z.1,z.2.2)),
+    (continuous_fst.comp continuous_snd).prodMk
+      (H.continuous.comp (continuous_fst.prodMk (continuous_snd.comp continuous_snd)))⟩
+
+include hH in
+lemma suspHomotopyRay_bottom (s : unitInterval) (a : A) :
+    suspHomotopyRay H (s,(0,a)) = south p (q a) := by
+  change meridian p (0,H (s,a)) = _
+  rw [ray_zero,hH]
+
+include hH in
+lemma suspHomotopyRay_top (s : unitInterval) (a : A) :
+    suspHomotopyRay H (s,(1,a)) = north p (q a) := by
+  change meridian p (1,H (s,a)) = _
+  rw [ray_one,hH]
+
+def suspHomotopy : (suspMap f p q hf).Homotopy (suspMap g p q hg) where
+  toContinuousMap := cylinderLift q q
+    ((south p).comp ⟨Prod.snd,continuous_snd⟩) (suspHomotopyRay H)
+    ((north p).comp ⟨Prod.snd,continuous_snd⟩)
+    (suspHomotopyRay_bottom H hH) (suspHomotopyRay_top H hH)
+  map_zero_left z := by
+    induction z using induction with
+    | hbase _ => rfl
+    | hray ta =>
+      change meridian p (ta.1,H (0,ta.2)) = meridian p (ta.1,f ta.2)
+      rw [HomotopyLike.map_zero_left H]
+    | hapex _ => rfl
+  map_one_left z := by
+    induction z using induction with
+    | hbase _ => rfl
+    | hray ta =>
+      change meridian p (ta.1,H (1,ta.2)) = meridian p (ta.1,g ta.2)
+      rw [HomotopyLike.map_one_left H]
+    | hapex _ => rfl
+
+lemma suspHomotopy_over (s : unitInterval) (z : Susp q) :
+    suspProjection p (suspHomotopy hf hg H hH (s,z)) = suspProjection q z := by
+  induction z using induction with
+  | hbase _ => rfl
+  | hray ta => exact hH s ta.2
+  | hapex _ => rfl
+
+end NearbyLagrangian.FiberCone
+
+end
+
+end OAI

lean/OAI/Topology/Homotopy/CubicalNullity.lean

@@ -1,263 +1,263 @@
-import Mathlib
-
-namespace OAI
-
-/-! Cubical free nullity and change of basepoint. -/
-
-noncomputable section
-universe u v w
-open Set ContinuousMap
-open scoped Topology Topology.Homotopy
-
-namespace NearbyLagrangian.CubicalFreeNullity
-variable {N : Type*} [Fintype N] [Nonempty N]
-
-def radius (t : N → unitInterval) : ℝ := ‖fun i => 2*(t i:ℝ)-1‖
-omit [Nonempty N] in
-lemma radius_nonneg (t : N → unitInterval) : 0 ≤ radius t := norm_nonneg _
-omit [Nonempty N] in
-lemma radius_le_one (t : N → unitInterval) : radius t ≤ 1 := by
-  apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ) ≤ 1)).mpr
-  intro i
-  rw [Real.norm_eq_abs,abs_le]
-  have hi₀ := (t i).property.1
-  have hi₁ := (t i).property.2
-  constructor <;> linarith
-omit [Nonempty N] in
-lemma continuous_radius : Continuous (radius (N:=N)) :=
-  (continuous_pi fun i => (continuous_const.mul
-    (continuous_subtype_val.comp (continuous_apply i))).sub continuous_const).norm
-omit [Nonempty N] in
-lemma radius_boundary {t : N → unitInterval} (ht : t ∈ Cube.boundary N) : radius t = 1 := by
-  apply le_antisymm (radius_le_one t)
-  obtain ⟨i,hi⟩ := ht
-  have hh := norm_le_pi_norm (fun i => 2*(t i:ℝ)-1) i
-  rcases hi with hi | hi
-  · simpa [radius,hi] using hh
-  · norm_num [hi] at hh
-    exact hh
-
-def collarTime (s : unitInterval) (t : N → unitInterval) : unitInterval :=
-  ⟨min (s:ℝ) (2*(1-radius t)),
-    le_min s.property.1 (by have := radius_le_one t; linarith),
-    (min_le_left _ _).trans s.property.2⟩
-omit [Nonempty N] in
-lemma continuous_collarTime : Continuous (fun a : unitInterval × (N → unitInterval) => collarTime a.1 a.2) :=
-  ((continuous_subtype_val.comp continuous_fst).min
-    (continuous_const.mul (continuous_const.sub (continuous_radius.comp continuous_snd)))).subtype_mk _
-omit [Nonempty N] in
-@[simp] lemma collarTime_zero (t : N → unitInterval) : collarTime 0 t = 0 := by
-  apply Subtype.ext
-  change min 0 (2*(1-radius t)) = 0
-  apply min_eq_left
-  have := radius_le_one t
-  linarith
-omit [Nonempty N] in
-lemma collarTime_boundary (s : unitInterval) {t : N → unitInterval} (ht : t ∈ Cube.boundary N) :
-    collarTime s t = 0 := by
-  apply Subtype.ext
-  change min (s:ℝ) (2*(1-radius t)) = 0
-  rw [radius_boundary ht]
-  simp only [sub_self,mul_zero,min_eq_right s.property.1]
-
-lemma denominator_pos (s : unitInterval) : 0 < 1-(s:ℝ)/2 := by
-  have := s.property.2
-  linarith
-
-def expand (s : unitInterval) (t : N → unitInterval) : N → unitInterval :=
-  fun i => projIcc 0 1 (zero_le_one' ℝ) (((t i:ℝ)-(s:ℝ)/4)/(1-(s:ℝ)/2))
-omit [Fintype N] [Nonempty N] in
-lemma continuous_expand : Continuous (fun a : unitInterval × (N → unitInterval) => expand a.1 a.2) := by
-  apply continuous_pi
-  intro i
-  apply continuous_projIcc.comp
-  exact ((continuous_subtype_val.comp ((continuous_apply i).comp continuous_snd)).sub
-    ((continuous_subtype_val.comp continuous_fst).div_const 4)).div
-      (continuous_const.sub ((continuous_subtype_val.comp continuous_fst).div_const 2))
-      (fun a => ne_of_gt (denominator_pos a.1))
-omit [Fintype N] [Nonempty N] in
-@[simp] lemma expand_zero (t : N → unitInterval) : expand 0 t = t := by
-  funext i
-  change projIcc 0 1 _ (((t i:ℝ)-(0:ℝ)/4)/(1-(0:ℝ)/2)) = t i
-  simp only [zero_div,sub_zero,div_one,projIcc_val]
-
-lemma expand_mem_boundary {s : unitInterval} {t : N → unitInterval}
-    (ht : 1-(s:ℝ)/2 ≤ radius t) : expand s t ∈ Cube.boundary N := by
-  obtain ⟨⟨i,hi⟩,_⟩ := IsGreatest.pi_norm (fun i => 2*(t i:ℝ)-1)
-  have habs : 1-(s:ℝ)/2 ≤ |2*(t i:ℝ)-1| := by
-    change ‖2*(t i:ℝ)-1‖ = radius t at hi
-    rw [Real.norm_eq_abs] at hi
-    exact ht.trans_eq hi.symm
-  rcases (le_abs.mp habs) with h | h
-  · refine ⟨i,Or.inr ?_⟩
-    apply projIcc_of_right_le
-    apply (le_div_iff₀ (denominator_pos s)).mpr
-    linarith
-  · refine ⟨i,Or.inl ?_⟩
-    apply projIcc_of_le_left
-    apply div_nonpos_of_nonpos_of_nonneg _ (denominator_pos s).le
-    linarith
-
-lemma expand_boundary (s : unitInterval) {t : N → unitInterval} (ht : t ∈ Cube.boundary N) :
-    expand s t ∈ Cube.boundary N := by
-  apply expand_mem_boundary
-  rw [radius_boundary ht]
-  have := s.property.1
-  linarith
-
-variable {X : Type*} [TopologicalSpace X]
-
-theorem null_of_moving_base {x y : X} (f : GenLoop N X x)
-    (H : f.1.Homotopy (ContinuousMap.const _ y))
-    (p : C(unitInterval,X))
-    (hp : ∀ s t, t ∈ Cube.boundary N → H (s,t) = p s) :
-    GenLoop.Homotopic f GenLoop.const := by
-  let z : N → unitInterval := fun _ => 0
-  have hz : z ∈ Cube.boundary N := ⟨Classical.choice inferInstance,Or.inl rfl⟩
-  have hp0 : p 0 = x := by
-    rw [← hp 0 z hz,H.apply_zero]
-    exact GenLoop.boundary f z hz
-  have hp1 : p 1 = y := by rw [← hp 1 z hz,H.apply_one]; rfl
-  let a : C(N → unitInterval,X) := p.comp ⟨fun t => collarTime 1 t,
-    continuous_collarTime.comp (continuous_const.prodMk continuous_id)⟩
-  have ha (t : N → unitInterval) : H (collarTime 1 t,expand 1 t) = a t := by
-    change H (collarTime 1 t,expand 1 t) = p (collarTime 1 t)
-    by_cases ht : collarTime 1 t = 1
-    · rw [ht,H.apply_one,hp1]; rfl
-    · apply hp
-      apply expand_mem_boundary
-      have hk : (collarTime 1 t:ℝ) < 1 := lt_of_le_of_ne (collarTime 1 t).property.2
-        (fun h => ht (Subtype.ext h))
-      change min 1 (2*(1-radius t)) < 1 at hk
-      have hr : 2*(1-radius t) < 1 := by simpa only [min_lt_iff,lt_self_iff_false,false_or] using hk
-      change 1-(1:ℝ)/2 ≤ radius t
-      linarith
-  let H₁ : f.1.HomotopyRel a (Cube.boundary N) := {
-    toFun := fun q => H (collarTime q.1 q.2,expand q.1 q.2)
-    continuous_toFun := H.continuous.comp (continuous_collarTime.prodMk continuous_expand)
-    map_zero_left := by intro t; simp only [collarTime_zero,expand_zero,H.apply_zero]
-    map_one_left := ha
-    prop' := by
-      intro s t ht
-      change H (collarTime s t,expand s t) = f t
-      rw [collarTime_boundary s ht,hp 0 _ (expand_boundary s ht),hp0,GenLoop.boundary f t ht] }
-  let H₂ : a.HomotopyRel (ContinuousMap.const _ x) (Cube.boundary N) := {
-    toFun := fun q => p ⟨(1-(q.1:ℝ))*(collarTime 1 q.2:ℝ),
-      mul_nonneg (by linarith [q.1.property.2]) (collarTime _ _).property.1,
-      (mul_le_of_le_one_left (collarTime _ _).property.1
-        (by linarith [q.1.property.1])).trans (collarTime _ _).property.2⟩
-    continuous_toFun := p.continuous.comp (Continuous.subtype_mk
-      ((continuous_const.sub (continuous_subtype_val.comp continuous_fst)).mul
-        (continuous_subtype_val.comp (continuous_collarTime.comp (continuous_const.prodMk continuous_snd)))) _)
-    map_zero_left := by
-      intro t
-      change p _ = p (collarTime 1 t)
-      congr 1
-      apply Subtype.ext
-      change (1-(0:ℝ))*(collarTime 1 t:ℝ) = (collarTime 1 t:ℝ)
-      ring
-    map_one_left := by
-      intro t
-      change p _ = x
-      convert hp0 using 1
-      congr 1
-      apply Subtype.ext
-      change (1-(1:ℝ))*(collarTime 1 t:ℝ) = 0
-      ring
-    prop' := by
-      intro s t ht
-      change p _ = p (collarTime 1 t)
-      rw [collarTime_boundary 1 ht]
-      congr 1
-      apply Subtype.ext
-      change (1-(s:ℝ))*(0:ℝ) = 0
-      ring }
-  exact ⟨H₁.trans H₂⟩
-
-end NearbyLagrangian.CubicalFreeNullity
-
-end
-
-noncomputable section
-universe u v w
-open scoped Classical Topology Topology.Homotopy
-
-namespace NearbyLagrangian.CubicalFreeNullity
-variable {N : Type*} [Fintype N] [Nonempty N]
-variable {X : Type*} [TopologicalSpace X] {x y : X}
-
-def outerTime (s : unitInterval) (t : N → unitInterval) : unitInterval :=
-  ⟨max 0 ((s:ℝ)-2*(1-radius t)),le_max_left _ _,by
-    apply max_le zero_le_one
-    have := radius_le_one t
-    linarith [s.property.2]⟩
-omit [Nonempty N] in
-lemma continuous_outerTime : Continuous (fun a : unitInterval × (N → unitInterval) => outerTime a.1 a.2) :=
-  ((continuous_const.max ((continuous_subtype_val.comp continuous_fst).sub
-    (continuous_const.mul (continuous_const.sub (continuous_radius.comp continuous_snd)))))).subtype_mk _
-
-omit [Nonempty N] in
-lemma outerTime_eq_zero {s : unitInterval} {t : N → unitInterval}
-    (h : radius t ≤ 1-(s:ℝ)/2) : outerTime s t = 0 := by
-  apply Subtype.ext
-  change max 0 ((s:ℝ)-2*(1-radius t)) = 0
-  apply max_eq_left
-  linarith
-
-omit [Nonempty N] in
-lemma outerTime_boundary (s : unitInterval) {t : N → unitInterval}
-    (h : t ∈ Cube.boundary N) : outerTime s t = s := by
-  apply Subtype.ext
-  change max 0 ((s:ℝ)-2*(1-radius t)) = s
-  rw [radius_boundary h]
-  simp only [sub_self,mul_zero,sub_zero,max_eq_right s.property.1]
-
-def rebaseValue (f : GenLoop N X x) (p : Path x y) (s : unitInterval)
-    (t : N → unitInterval) : X :=
-  if radius t ≤ 1-(s:ℝ)/2 then f (expand s t) else p (outerTime s t)
-
-lemma continuous_rebaseValue (f : GenLoop N X x) (p : Path x y) :
-    Continuous (fun q : unitInterval × (N → unitInterval) => rebaseValue f p q.1 q.2) := by
-  apply (f.1.continuous.comp continuous_expand).if_le
-    (p.continuous.comp continuous_outerTime) (continuous_radius.comp continuous_snd)
-    (continuous_const.sub ((continuous_subtype_val.comp continuous_fst).div_const 2))
-  intro q h
-  change f (expand q.1 q.2) = p (outerTime q.1 q.2)
-  change radius q.2 = 1-(q.1:ℝ)/2 at h
-  rw [GenLoop.boundary f _ (expand_mem_boundary h.ge),outerTime_eq_zero h.le,p.source]
-
-omit [Nonempty N] in
-@[simp] lemma rebaseValue_zero (f : GenLoop N X x) (p : Path x y)
-    (t : N → unitInterval) : rebaseValue f p 0 t = f t := by
-  rw [rebaseValue,ite_eq_left (by simpa using radius_le_one t),expand_zero]
-
-omit [Nonempty N] in
-lemma rebaseValue_boundary (f : GenLoop N X x) (p : Path x y)
-    (s : unitInterval) {t : N → unitInterval} (ht : t ∈ Cube.boundary N) :
-    rebaseValue f p s t = p s := by
-  rw [rebaseValue]
-  split_ifs with h
-  · have hs : s = 0 := by
-      apply Subtype.ext
-      change (s:ℝ) = 0
-      rw [radius_boundary ht] at h
-      linarith [s.property.1]
-    rw [hs,expand_zero,GenLoop.boundary f t ht,p.source]
-  · rw [outerTime_boundary s ht]
-
-def rebase (f : GenLoop N X x) (p : Path x y) : GenLoop N X y :=
-  ⟨⟨fun t => rebaseValue f p 1 t,
-    (continuous_rebaseValue f p).comp (continuous_const.prodMk continuous_id)⟩,
-    fun _t ht => (rebaseValue_boundary f p 1 ht).trans p.target⟩
-
-def rebaseHomotopy (f : GenLoop N X x) (p : Path x y) : f.1.Homotopy (rebase f p).1 where
-  toFun q := rebaseValue f p q.1 q.2
-  continuous_toFun := continuous_rebaseValue f p
-  map_zero_left := rebaseValue_zero f p
-  map_one_left _ := rfl
-
-end NearbyLagrangian.CubicalFreeNullity
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Cubical free nullity and change of basepoint. -/
+
+noncomputable section
+universe u v w
+open Set ContinuousMap
+open scoped Topology Topology.Homotopy
+
+namespace NearbyLagrangian.CubicalFreeNullity
+variable {N : Type*} [Fintype N] [Nonempty N]
+
+def radius (t : N → unitInterval) : ℝ := ‖fun i => 2*(t i:ℝ)-1‖
+omit [Nonempty N] in
+lemma radius_nonneg (t : N → unitInterval) : 0 ≤ radius t := norm_nonneg _
+omit [Nonempty N] in
+lemma radius_le_one (t : N → unitInterval) : radius t ≤ 1 := by
+  apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ) ≤ 1)).mpr
+  intro i
+  rw [Real.norm_eq_abs,abs_le]
+  have hi₀ := (t i).property.1
+  have hi₁ := (t i).property.2
+  constructor <;> linarith
+omit [Nonempty N] in
+lemma continuous_radius : Continuous (radius (N:=N)) :=
+  (continuous_pi fun i => (continuous_const.mul
+    (continuous_subtype_val.comp (continuous_apply i))).sub continuous_const).norm
+omit [Nonempty N] in
+lemma radius_boundary {t : N → unitInterval} (ht : t ∈ Cube.boundary N) : radius t = 1 := by
+  apply le_antisymm (radius_le_one t)
+  obtain ⟨i,hi⟩ := ht
+  have hh := norm_le_pi_norm (fun i => 2*(t i:ℝ)-1) i
+  rcases hi with hi | hi
+  · simpa [radius,hi] using hh
+  · norm_num [hi] at hh
+    exact hh
+
+def collarTime (s : unitInterval) (t : N → unitInterval) : unitInterval :=
+  ⟨min (s:ℝ) (2*(1-radius t)),
+    le_min s.property.1 (by have := radius_le_one t; linarith),
+    (min_le_left _ _).trans s.property.2⟩
+omit [Nonempty N] in
+lemma continuous_collarTime : Continuous (fun a : unitInterval × (N → unitInterval) => collarTime a.1 a.2) :=
+  ((continuous_subtype_val.comp continuous_fst).min
+    (continuous_const.mul (continuous_const.sub (continuous_radius.comp continuous_snd)))).subtype_mk _
+omit [Nonempty N] in
+@[simp] lemma collarTime_zero (t : N → unitInterval) : collarTime 0 t = 0 := by
+  apply Subtype.ext
+  change min 0 (2*(1-radius t)) = 0
+  apply min_eq_left
+  have := radius_le_one t
+  linarith
+omit [Nonempty N] in
+lemma collarTime_boundary (s : unitInterval) {t : N → unitInterval} (ht : t ∈ Cube.boundary N) :
+    collarTime s t = 0 := by
+  apply Subtype.ext
+  change min (s:ℝ) (2*(1-radius t)) = 0
+  rw [radius_boundary ht]
+  simp only [sub_self,mul_zero,min_eq_right s.property.1]
+
+lemma denominator_pos (s : unitInterval) : 0 < 1-(s:ℝ)/2 := by
+  have := s.property.2
+  linarith
+
+def expand (s : unitInterval) (t : N → unitInterval) : N → unitInterval :=
+  fun i => projIcc 0 1 (zero_le_one' ℝ) (((t i:ℝ)-(s:ℝ)/4)/(1-(s:ℝ)/2))
+omit [Fintype N] [Nonempty N] in
+lemma continuous_expand : Continuous (fun a : unitInterval × (N → unitInterval) => expand a.1 a.2) := by
+  apply continuous_pi
+  intro i
+  apply continuous_projIcc.comp
+  exact ((continuous_subtype_val.comp ((continuous_apply i).comp continuous_snd)).sub
+    ((continuous_subtype_val.comp continuous_fst).div_const 4)).div
+      (continuous_const.sub ((continuous_subtype_val.comp continuous_fst).div_const 2))
+      (fun a => ne_of_gt (denominator_pos a.1))
+omit [Fintype N] [Nonempty N] in
+@[simp] lemma expand_zero (t : N → unitInterval) : expand 0 t = t := by
+  funext i
+  change projIcc 0 1 _ (((t i:ℝ)-(0:ℝ)/4)/(1-(0:ℝ)/2)) = t i
+  simp only [zero_div,sub_zero,div_one,projIcc_val]
+
+lemma expand_mem_boundary {s : unitInterval} {t : N → unitInterval}
+    (ht : 1-(s:ℝ)/2 ≤ radius t) : expand s t ∈ Cube.boundary N := by
+  obtain ⟨⟨i,hi⟩,_⟩ := IsGreatest.pi_norm (fun i => 2*(t i:ℝ)-1)
+  have habs : 1-(s:ℝ)/2 ≤ |2*(t i:ℝ)-1| := by
+    change ‖2*(t i:ℝ)-1‖ = radius t at hi
+    rw [Real.norm_eq_abs] at hi
+    exact ht.trans_eq hi.symm
+  rcases (le_abs.mp habs) with h | h
+  · refine ⟨i,Or.inr ?_⟩
+    apply projIcc_of_right_le
+    apply (le_div_iff₀ (denominator_pos s)).mpr
+    linarith
+  · refine ⟨i,Or.inl ?_⟩
+    apply projIcc_of_le_left
+    apply div_nonpos_of_nonpos_of_nonneg _ (denominator_pos s).le
+    linarith
+
+lemma expand_boundary (s : unitInterval) {t : N → unitInterval} (ht : t ∈ Cube.boundary N) :
+    expand s t ∈ Cube.boundary N := by
+  apply expand_mem_boundary
+  rw [radius_boundary ht]
+  have := s.property.1
+  linarith
+
+variable {X : Type*} [TopologicalSpace X]
+
+theorem null_of_moving_base {x y : X} (f : GenLoop N X x)
+    (H : f.1.Homotopy (ContinuousMap.const _ y))
+    (p : C(unitInterval,X))
+    (hp : ∀ s t, t ∈ Cube.boundary N → H (s,t) = p s) :
+    GenLoop.Homotopic f GenLoop.const := by
+  let z : N → unitInterval := fun _ => 0
+  have hz : z ∈ Cube.boundary N := ⟨Classical.choice inferInstance,Or.inl rfl⟩
+  have hp0 : p 0 = x := by
+    rw [← hp 0 z hz,H.apply_zero]
+    exact GenLoop.boundary f z hz
+  have hp1 : p 1 = y := by rw [← hp 1 z hz,H.apply_one]; rfl
+  let a : C(N → unitInterval,X) := p.comp ⟨fun t => collarTime 1 t,
+    continuous_collarTime.comp (continuous_const.prodMk continuous_id)⟩
+  have ha (t : N → unitInterval) : H (collarTime 1 t,expand 1 t) = a t := by
+    change H (collarTime 1 t,expand 1 t) = p (collarTime 1 t)
+    by_cases ht : collarTime 1 t = 1
+    · rw [ht,H.apply_one,hp1]; rfl
+    · apply hp
+      apply expand_mem_boundary
+      have hk : (collarTime 1 t:ℝ) < 1 := lt_of_le_of_ne (collarTime 1 t).property.2
+        (fun h => ht (Subtype.ext h))
+      change min 1 (2*(1-radius t)) < 1 at hk
+      have hr : 2*(1-radius t) < 1 := by simpa only [min_lt_iff,lt_self_iff_false,false_or] using hk
+      change 1-(1:ℝ)/2 ≤ radius t
+      linarith
+  let H₁ : f.1.HomotopyRel a (Cube.boundary N) := {
+    toFun := fun q => H (collarTime q.1 q.2,expand q.1 q.2)
+    continuous_toFun := H.continuous.comp (continuous_collarTime.prodMk continuous_expand)
+    map_zero_left := by intro t; simp only [collarTime_zero,expand_zero,H.apply_zero]
+    map_one_left := ha
+    prop' := by
+      intro s t ht
+      change H (collarTime s t,expand s t) = f t
+      rw [collarTime_boundary s ht,hp 0 _ (expand_boundary s ht),hp0,GenLoop.boundary f t ht] }
+  let H₂ : a.HomotopyRel (ContinuousMap.const _ x) (Cube.boundary N) := {
+    toFun := fun q => p ⟨(1-(q.1:ℝ))*(collarTime 1 q.2:ℝ),
+      mul_nonneg (by linarith [q.1.property.2]) (collarTime _ _).property.1,
+      (mul_le_of_le_one_left (collarTime _ _).property.1
+        (by linarith [q.1.property.1])).trans (collarTime _ _).property.2⟩
+    continuous_toFun := p.continuous.comp (Continuous.subtype_mk
+      ((continuous_const.sub (continuous_subtype_val.comp continuous_fst)).mul
+        (continuous_subtype_val.comp (continuous_collarTime.comp (continuous_const.prodMk continuous_snd)))) _)
+    map_zero_left := by
+      intro t
+      change p _ = p (collarTime 1 t)
+      congr 1
+      apply Subtype.ext
+      change (1-(0:ℝ))*(collarTime 1 t:ℝ) = (collarTime 1 t:ℝ)
+      ring
+    map_one_left := by
+      intro t
+      change p _ = x
+      convert hp0 using 1
+      congr 1
+      apply Subtype.ext
+      change (1-(1:ℝ))*(collarTime 1 t:ℝ) = 0
+      ring
+    prop' := by
+      intro s t ht
+      change p _ = p (collarTime 1 t)
+      rw [collarTime_boundary 1 ht]
+      congr 1
+      apply Subtype.ext
+      change (1-(s:ℝ))*(0:ℝ) = 0
+      ring }
+  exact ⟨H₁.trans H₂⟩
+
+end NearbyLagrangian.CubicalFreeNullity
+
+end
+
+noncomputable section
+universe u v w
+open scoped Classical Topology Topology.Homotopy
+
+namespace NearbyLagrangian.CubicalFreeNullity
+variable {N : Type*} [Fintype N] [Nonempty N]
+variable {X : Type*} [TopologicalSpace X] {x y : X}
+
+def outerTime (s : unitInterval) (t : N → unitInterval) : unitInterval :=
+  ⟨max 0 ((s:ℝ)-2*(1-radius t)),le_max_left _ _,by
+    apply max_le zero_le_one
+    have := radius_le_one t
+    linarith [s.property.2]⟩
+omit [Nonempty N] in
+lemma continuous_outerTime : Continuous (fun a : unitInterval × (N → unitInterval) => outerTime a.1 a.2) :=
+  ((continuous_const.max ((continuous_subtype_val.comp continuous_fst).sub
+    (continuous_const.mul (continuous_const.sub (continuous_radius.comp continuous_snd)))))).subtype_mk _
+
+omit [Nonempty N] in
+lemma outerTime_eq_zero {s : unitInterval} {t : N → unitInterval}
+    (h : radius t ≤ 1-(s:ℝ)/2) : outerTime s t = 0 := by
+  apply Subtype.ext
+  change max 0 ((s:ℝ)-2*(1-radius t)) = 0
+  apply max_eq_left
+  linarith
+
+omit [Nonempty N] in
+lemma outerTime_boundary (s : unitInterval) {t : N → unitInterval}
+    (h : t ∈ Cube.boundary N) : outerTime s t = s := by
+  apply Subtype.ext
+  change max 0 ((s:ℝ)-2*(1-radius t)) = s
+  rw [radius_boundary h]
+  simp only [sub_self,mul_zero,sub_zero,max_eq_right s.property.1]
+
+def rebaseValue (f : GenLoop N X x) (p : Path x y) (s : unitInterval)
+    (t : N → unitInterval) : X :=
+  if radius t ≤ 1-(s:ℝ)/2 then f (expand s t) else p (outerTime s t)
+
+lemma continuous_rebaseValue (f : GenLoop N X x) (p : Path x y) :
+    Continuous (fun q : unitInterval × (N → unitInterval) => rebaseValue f p q.1 q.2) := by
+  apply (f.1.continuous.comp continuous_expand).if_le
+    (p.continuous.comp continuous_outerTime) (continuous_radius.comp continuous_snd)
+    (continuous_const.sub ((continuous_subtype_val.comp continuous_fst).div_const 2))
+  intro q h
+  change f (expand q.1 q.2) = p (outerTime q.1 q.2)
+  change radius q.2 = 1-(q.1:ℝ)/2 at h
+  rw [GenLoop.boundary f _ (expand_mem_boundary h.ge),outerTime_eq_zero h.le,p.source]
+
+omit [Nonempty N] in
+@[simp] lemma rebaseValue_zero (f : GenLoop N X x) (p : Path x y)
+    (t : N → unitInterval) : rebaseValue f p 0 t = f t := by
+  rw [rebaseValue,ite_eq_left (by simpa using radius_le_one t),expand_zero]
+
+omit [Nonempty N] in
+lemma rebaseValue_boundary (f : GenLoop N X x) (p : Path x y)
+    (s : unitInterval) {t : N → unitInterval} (ht : t ∈ Cube.boundary N) :
+    rebaseValue f p s t = p s := by
+  rw [rebaseValue]
+  split_ifs with h
+  · have hs : s = 0 := by
+      apply Subtype.ext
+      change (s:ℝ) = 0
+      rw [radius_boundary ht] at h
+      linarith [s.property.1]
+    rw [hs,expand_zero,GenLoop.boundary f t ht,p.source]
+  · rw [outerTime_boundary s ht]
+
+def rebase (f : GenLoop N X x) (p : Path x y) : GenLoop N X y :=
+  ⟨⟨fun t => rebaseValue f p 1 t,
+    (continuous_rebaseValue f p).comp (continuous_const.prodMk continuous_id)⟩,
+    fun _t ht => (rebaseValue_boundary f p 1 ht).trans p.target⟩
+
+def rebaseHomotopy (f : GenLoop N X x) (p : Path x y) : f.1.Homotopy (rebase f p).1 where
+  toFun q := rebaseValue f p q.1 q.2
+  continuous_toFun := continuous_rebaseValue f p
+  map_zero_left := rebaseValue_zero f p
+  map_one_left _ := rfl
+
+end NearbyLagrangian.CubicalFreeNullity
+
+end
+
+end OAI

lean/OAI/Topology/Homotopy/CubicalSpheres.lean

@@ -1,341 +1,341 @@
-import Mathlib
-import OAI.Topology.Homotopy.CubicalNullity
-import OAI.Topology.Homotopy.OpenCollapse
-
-namespace OAI
-
-/-! Cubical and spherical representatives of homotopy classes. -/
-
-noncomputable section
-universe u v w
-open Set ContinuousMap Topology
-open scoped OnePoint Topology Topology.Homotopy
-
-namespace NearbyLagrangian.CubicalSphere
-variable (N : Type*) [Fintype N] [Nonempty N]
-
-def interiorCube : Set (N → unitInterval) := {t | ∀ i, 0 < (t i:ℝ) ∧ (t i:ℝ) < 1}
-omit [Fintype N] [Nonempty N] in
-lemma interiorCube_eq : interiorCube N = (Cube.boundary N)ᶜ := by
-  ext t
-  simp only [interiorCube,mem_ofPred_eq,Set.mem_compl_iff,Cube.boundary,not_exists,not_or]
-  constructor
-  · intro ht i
-    constructor
-    · intro h; have := (ht i).1; simp [h] at this
-    · intro h; have := (ht i).2; simp [h] at this
-  · intro ht i
-    constructor
-    · exact lt_of_le_of_ne (t i).property.1 (fun h => (ht i).1 (Subtype.ext h.symm))
-    · exact lt_of_le_of_ne (t i).property.2 (fun h => (ht i).2 (Subtype.ext h))
-
-omit [Nonempty N] in
-lemma isOpen_interiorCube : IsOpen (interiorCube N) := by
-  have he : interiorCube N = ⋂ i, {t : N → unitInterval | 0 < (t i:ℝ) ∧ (t i:ℝ) < 1} := by
-    ext t; simp [interiorCube]
-  rw [he]
-  apply isOpen_iInter_of_finite
-  intro i
-  exact isOpen_Ioo.preimage (continuous_subtype_val.comp (continuous_apply i))
-
-omit [Fintype N] in
-lemma complement_nonempty : (interiorCube N)ᶜ.Nonempty := by
-  refine ⟨fun _ => 0,?_⟩
-  rw [interiorCube_eq,compl_compl]
-  exact ⟨Classical.choice inferInstance,Or.inl rfl⟩
-
-def interiorHomeomorph : interiorCube N ≃ₜ (N → Ioo (0:ℝ) 1) where
-  toFun t i := ⟨t.1 i,(t.property i)⟩
-  invFun t := ⟨fun i => ⟨(t i:ℝ),(t i).property.1.le,(t i).property.2.le⟩,
-    fun i => (t i).property⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  continuous_toFun := by
-    apply continuous_pi
-    intro i
-    exact (continuous_subtype_val.comp ((continuous_apply i).comp continuous_subtype_val)).subtype_mk _
-  continuous_invFun := by
-    apply Continuous.subtype_mk
-    apply continuous_pi
-    intro i
-    exact (continuous_subtype_val.comp (continuous_apply i)).subtype_mk _
-
-def intervalHomeomorph : Ioo (0:ℝ) 1 ≃ₜ Ioo (-1:ℝ) 1 where
-  toFun x := ⟨2*(x:ℝ)-1,by have h := x.property; constructor <;> linarith [h.1,h.2]⟩
-  invFun y := ⟨((y:ℝ)+1)/2,by have h := y.property; constructor <;> linarith [h.1,h.2]⟩
-  left_inv x := Subtype.ext (by dsimp; ring)
-  right_inv y := Subtype.ext (by dsimp; ring)
-  continuous_toFun := by
-    apply Continuous.subtype_mk
-    exact (continuous_const.mul continuous_subtype_val).sub continuous_const
-  continuous_invFun := by
-    apply Continuous.subtype_mk
-    exact (continuous_subtype_val.add continuous_const).div_const 2
-
-def interiorEuclidean : interiorCube N ≃ₜ (N → ℝ) :=
-  (interiorHomeomorph N).trans (Homeomorph.piCongrRight fun _ =>
-    intervalHomeomorph.trans (orderIsoIooNegOneOne ℝ).toHomeomorph.symm)
-
-def compactificationSphere (n : ℕ) : OnePoint (interiorCube (Fin (n+1))) ≃ₜ
-    Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1 :=
-  (interiorEuclidean (Fin (n+1))).onePointCongr.trans
-    (onePointEquivSphereOfFinrankEq (by simp))
-
-def spherePoint (n : ℕ) : Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1 :=
-  compactificationSphere n ∞
-
-def sphereQuotient (n : ℕ) : C((Fin (n+1) → unitInterval),
-    Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1) :=
-  ⟨fun t => compactificationSphere n (OpenCollapse.collapse (interiorCube _) t),
-    (compactificationSphere n).continuous.comp
-      (OpenCollapse.continuous_collapse (isOpen_interiorCube _))⟩
-
-lemma sphereQuotient_boundary (n : ℕ) {t : Fin (n+1) → unitInterval}
-    (ht : t ∈ Cube.boundary _) : sphereQuotient n t = spherePoint n := by
-  change compactificationSphere n _ = compactificationSphere n ∞
-  rw [OpenCollapse.collapse_of_notMem]
-  simpa only [interiorCube_eq,mem_compl_iff,not_not] using ht
-
-lemma sphereQuotient_surjective (n : ℕ) : Function.Surjective (sphereQuotient n) :=
-  (compactificationSphere n).surjective.comp
-    (OpenCollapse.collapse_surjective (complement_nonempty _))
-
-variable {X : Type*} [TopologicalSpace X] {x : X}
-
-def sphereMap {n : ℕ} (f : GenLoop (Fin (n+1)) X x) :
-    C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X) :=
-  (OpenCollapse.descend (isOpen_interiorCube _) (complement_nonempty _) f.1 x
-      (fun t ht => GenLoop.boundary f t (by simpa only [interiorCube_eq,mem_compl_iff,not_not] using ht))).comp
-    ⟨(compactificationSphere n).symm,(compactificationSphere n).symm.continuous⟩
-
-@[simp] lemma sphereMap_quotient {n : ℕ} (f : GenLoop (Fin (n+1)) X x)
-    (t : Fin (n+1) → unitInterval) : sphereMap f (sphereQuotient n t) = f t := by
-  simp only [sphereMap,ContinuousMap.comp_apply,ContinuousMap.coe_mk,sphereQuotient,
-    Homeomorph.symm_apply_apply,OpenCollapse.descend_collapse]
-  rfl
-
-@[simp] lemma sphereMap_point {n : ℕ} (f : GenLoop (Fin (n+1)) X x) :
-    sphereMap f (spherePoint n) = x := by
-  simp only [sphereMap,ContinuousMap.comp_apply,ContinuousMap.coe_mk,spherePoint,
-    Homeomorph.symm_apply_apply]
-  rfl
-
-theorem sphereMap_null_implies_cube_null {n : ℕ} (f : GenLoop (Fin (n+1)) X x)
-    (hf : (sphereMap f).Nullhomotopic) : GenLoop.Homotopic f GenLoop.const := by
-  obtain ⟨y,⟨H⟩⟩ := hf
-  let H' : f.1.Homotopy (ContinuousMap.const _ y) := {
-    toFun := fun q => H (q.1,sphereQuotient n q.2)
-    continuous_toFun := H.continuous.comp (continuous_fst.prodMk ((sphereQuotient n).continuous.comp continuous_snd))
-    map_zero_left := fun t => (H.apply_zero _).trans (sphereMap_quotient f t)
-    map_one_left := fun t => H.apply_one _ }
-  let p : C(unitInterval,X) :=
-    ⟨fun s => H (s,spherePoint n),H.continuous.comp (continuous_id.prodMk continuous_const)⟩
-  apply CubicalFreeNullity.null_of_moving_base f H' p
-  intro s t ht
-  change H (s,sphereQuotient n t) = H (s,spherePoint n)
-  rw [sphereQuotient_boundary n ht]
-
-theorem sphereMap_null_implies_class_one {n : ℕ} (f : GenLoop (Fin (n+1)) X x)
-    (hf : (sphereMap f).Nullhomotopic) :
-    (⟦f⟧ : HomotopyGroup.Pi (n+1) X x) = (1 : HomotopyGroup.Pi (n+1) X x) := by
-  rw [HomotopyGroup.one_def]
-  exact Quotient.sound (sphereMap_null_implies_cube_null f hf)
-
-end NearbyLagrangian.CubicalSphere
-
-end
-
-noncomputable section
-universe u v w
-open Set ContinuousMap Topology
-open scoped OnePoint Topology Topology.Homotopy
-
-namespace NearbyLagrangian.CubicalSphere
-variable {X : Type*} [TopologicalSpace X] {x : X}
-
-def sphereHomotopy {n : ℕ} {f g : GenLoop (Fin (n+1)) X x}
-    (H : f.1.HomotopyRel g.1 (Cube.boundary _)) : (sphereMap f).Homotopy (sphereMap g) where
-  toFun := fun q => ((compactificationSphere n).symm q.2).elim x
-    (fun t => H (q.1,t))
-  continuous_toFun := by
-    let F : C(unitInterval × (Fin (n+1) → unitInterval),X) := ⟨H,H.continuous⟩
-    have hF : ∀ s t, t ∉ interiorCube (Fin (n+1)) → F (s,t) = x := by
-      intro s t ht
-      have ht' : t ∈ Cube.boundary _ := by
-        simpa only [interiorCube_eq,mem_compl_iff,not_not] using ht
-      exact (H.eq_fst s ht').trans (GenLoop.boundary f t ht')
-    exact (OpenCollapse.continuous_descend_param (isOpen_interiorCube _) (complement_nonempty _)
-      F (const _ x) hF).comp (continuous_fst.prodMk
-        ((compactificationSphere n).symm.continuous.comp continuous_snd))
-  map_zero_left := by
-    intro u
-    change ((compactificationSphere n).symm u).elim x (fun t => H (0,t)) =
-      ((compactificationSphere n).symm u).elim x (fun t => f t)
-    generalize (compactificationSphere n).symm u = z
-    induction z using OnePoint.rec with
-    | infty => rfl
-    | coe t => exact H.apply_zero t
-  map_one_left := by
-    intro u
-    change ((compactificationSphere n).symm u).elim x (fun t => H (1,t)) =
-      ((compactificationSphere n).symm u).elim x (fun t => g t)
-    generalize (compactificationSphere n).symm u = z
-    induction z using OnePoint.rec with
-    | infty => rfl
-    | coe t => exact H.apply_one t
-
-@[simp] theorem sphereMap_const (n : ℕ) : sphereMap (GenLoop.const : GenLoop (Fin (n+1)) X x) =
-    ContinuousMap.const _ x := by
-  ext u
-  obtain ⟨t,rfl⟩ := sphereQuotient_surjective n u
-  exact sphereMap_quotient GenLoop.const t
-
-theorem sphereMap_null_iff_cube_null {n : ℕ} (f : GenLoop (Fin (n+1)) X x) :
-    (sphereMap f).Nullhomotopic ↔ GenLoop.Homotopic f GenLoop.const := by
-  constructor
-  · exact sphereMap_null_implies_cube_null f
-  · rintro ⟨H⟩
-    refine ⟨x,?_⟩
-    rw [← sphereMap_const]
-    exact ⟨sphereHomotopy H⟩
-
-theorem sphereMap_null_iff_class_one {n : ℕ} (f : GenLoop (Fin (n+1)) X x) :
-    (sphereMap f).Nullhomotopic ↔
-      (⟦f⟧ : HomotopyGroup.Pi (n+1) X x) = (1 : HomotopyGroup.Pi (n+1) X x) := by
-  rw [sphereMap_null_iff_cube_null,HomotopyGroup.one_def,Quotient.eq]
-  rfl
-
-end NearbyLagrangian.CubicalSphere
-
-end
-
-noncomputable section
-universe u v w
-
-
-namespace NearbyLagrangian.SphereConnectivity
-open scoped Topology Topology.Homotopy
-
-def genLoopMap {N A B : Type*} [TopologicalSpace A] [TopologicalSpace B]
-    (e : C(A,B)) {x : A} (f : GenLoop N A x) : GenLoop N B (e x) :=
-  ⟨e.comp f.1,fun t ht => congrArg e (GenLoop.boundary f t ht)⟩
-lemma genLoopMap_homotopic {N A B : Type*} [TopologicalSpace A] [TopologicalSpace B]
-    (e : C(A,B)) {x : A} {f g : GenLoop N A x} (h : GenLoop.Homotopic f g) :
-    GenLoop.Homotopic (genLoopMap e f) (genLoopMap e g) := by
-  obtain ⟨H⟩ := h
-  exact ⟨H.compContinuousMap e⟩
-def homotopyGroupMap {n : ℕ} {A B : Type*} [TopologicalSpace A] [TopologicalSpace B]
-    (e : C(A,B)) (x : A) : HomotopyGroup.Pi n A x → HomotopyGroup.Pi n B (e x) :=
-  Quotient.map (genLoopMap e) (fun _ _ h => genLoopMap_homotopic e h)
-end NearbyLagrangian.SphereConnectivity
-
-end
-
-noncomputable section
-universe u v w
-
-
-namespace NearbyLagrangian.CubicalSphere
-open Set ContinuousMap SphereConnectivity
-open scoped Topology Topology.Homotopy
-variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {x : X}
-
-def sphereLoop {n : ℕ}
-    (a : C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X)) :
-    GenLoop (Fin (n+1)) X (a (spherePoint n)) :=
-  ⟨a.comp (sphereQuotient n),fun _t ht => congrArg a (sphereQuotient_boundary n ht)⟩
-
-@[simp] theorem sphereMap_sphereLoop {n : ℕ}
-    (a : C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X)) :
-    sphereMap (sphereLoop a) = a := by
-  ext u
-  obtain ⟨t,rfl⟩ := sphereQuotient_surjective n u
-  rw [sphereMap_quotient]
-  rfl
-
-@[simp] theorem sphereMap_natural {n : ℕ} (e : C(X,Y))
-    (f : GenLoop (Fin (n+1)) X x) :
-    sphereMap (genLoopMap e f) = e.comp (sphereMap f) := by
-  ext u
-  obtain ⟨t,rfl⟩ := sphereQuotient_surjective n u
-  rw [sphereMap_quotient,ContinuousMap.comp_apply,sphereMap_quotient]
-  rfl
-
-theorem null_iff_class_one {n : ℕ}
-    (a : C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X)) :
-    a.Nullhomotopic ↔
-      (⟦sphereLoop a⟧ : HomotopyGroup.Pi (n+1) X (a (spherePoint n))) =
-        (1 : HomotopyGroup.Pi (n+1) X (a (spherePoint n))) := by
-  simpa only [sphereMap_sphereLoop] using sphereMap_null_iff_class_one (sphereLoop a)
-
-theorem comp_null_iff_mapped_class_one {n : ℕ} (e : C(X,Y))
-    (a : C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X)) :
-    (e.comp a).Nullhomotopic ↔
-      homotopyGroupMap (n:=n+1) e (a (spherePoint n)) ⟦sphereLoop a⟧ =
-        (1 : HomotopyGroup.Pi (n+1) Y (e (a (spherePoint n)))) := by
-  change (e.comp a).Nullhomotopic ↔
-    (⟦genLoopMap e (sphereLoop a)⟧ : HomotopyGroup.Pi (n+1) Y (e (a (spherePoint n)))) = _
-  simpa only [sphereMap_natural,sphereMap_sphereLoop] using
-    sphereMap_null_iff_class_one (genLoopMap e (sphereLoop a))
-
-end NearbyLagrangian.CubicalSphere
-
-end
-
-noncomputable section
-universe u v w
-open scoped Classical Topology Topology.Homotopy
-
-namespace NearbyLagrangian.CubicalSphere
-open Set ContinuousMap
-open scoped OnePoint
-variable {X : Type*} [TopologicalSpace X] {x y : X}
-
-def sphereMovingHomotopy {n : ℕ} {f : GenLoop (Fin (n+1)) X x}
-    {g : GenLoop (Fin (n+1)) X y} (H : f.1.Homotopy g.1)
-    (p : Path x y) (hp : ∀ s t, t ∈ Cube.boundary (Fin (n+1)) → H (s,t) = p s) :
-    (sphereMap f).Homotopy (sphereMap g) where
-  toFun q := ((compactificationSphere n).symm q.2).elim (p q.1) (fun t => H (q.1,t))
-  continuous_toFun := by
-    let F : C(unitInterval × (Fin (n+1) → unitInterval),X) := ⟨H,H.continuous⟩
-    have hF (s : unitInterval) (t : Fin (n+1) → unitInterval)
-        (ht : t ∉ interiorCube (Fin (n+1))) : F (s,t) = p s := by
-      apply hp
-      simpa only [interiorCube_eq,mem_compl_iff,not_not] using ht
-    exact (OpenCollapse.continuous_descend_param (isOpen_interiorCube _) (complement_nonempty _)
-      F p.toContinuousMap hF).comp (continuous_fst.prodMk
-        ((compactificationSphere n).symm.continuous.comp continuous_snd))
-  map_zero_left := by
-    intro u
-    change ((compactificationSphere n).symm u).elim (p 0) (fun t => H (0,t)) =
-      ((compactificationSphere n).symm u).elim x (fun t => f t)
-    rw [p.source]
-    generalize (compactificationSphere n).symm u = z
-    induction z using OnePoint.rec with
-    | infty => rfl
-    | coe t => exact H.apply_zero t
-  map_one_left := by
-    intro u
-    change ((compactificationSphere n).symm u).elim (p 1) (fun t => H (1,t)) =
-      ((compactificationSphere n).symm u).elim y (fun t => g t)
-    rw [p.target]
-    generalize (compactificationSphere n).symm u = z
-    induction z using OnePoint.rec with
-    | infty => rfl
-    | coe t => exact H.apply_one t
-
-theorem exists_based_representative [PathConnectedSpace X] {n : ℕ}
-    (a : C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X)) (x : X) :
-    ∃ f : GenLoop (Fin (n+1)) X x, a.Homotopic (sphereMap f) := by
-  obtain ⟨p⟩ := PathConnectedSpace.joined (a (spherePoint n)) x
-  refine ⟨CubicalFreeNullity.rebase (sphereLoop a) p,?_⟩
-  have h := sphereMovingHomotopy (CubicalFreeNullity.rebaseHomotopy (sphereLoop a) p) p
-    (fun s _t ht => CubicalFreeNullity.rebaseValue_boundary (sphereLoop a) p s ht)
-  simpa only [sphereMap_sphereLoop] using (show (sphereMap (sphereLoop a)).Homotopic
-    (sphereMap (CubicalFreeNullity.rebase (sphereLoop a) p)) from ⟨h⟩)
-
-end NearbyLagrangian.CubicalSphere
-
-end
-
-end OAI
+import Mathlib
+import OAI.Topology.Homotopy.CubicalNullity
+import OAI.Topology.Homotopy.OpenCollapse
+
+namespace OAI
+
+/-! Cubical and spherical representatives of homotopy classes. -/
+
+noncomputable section
+universe u v w
+open Set ContinuousMap Topology
+open scoped OnePoint Topology Topology.Homotopy
+
+namespace NearbyLagrangian.CubicalSphere
+variable (N : Type*) [Fintype N] [Nonempty N]
+
+def interiorCube : Set (N → unitInterval) := {t | ∀ i, 0 < (t i:ℝ) ∧ (t i:ℝ) < 1}
+omit [Fintype N] [Nonempty N] in
+lemma interiorCube_eq : interiorCube N = (Cube.boundary N)ᶜ := by
+  ext t
+  simp only [interiorCube,mem_ofPred_eq,Set.mem_compl_iff,Cube.boundary,not_exists,not_or]
+  constructor
+  · intro ht i
+    constructor
+    · intro h; have := (ht i).1; simp [h] at this
+    · intro h; have := (ht i).2; simp [h] at this
+  · intro ht i
+    constructor
+    · exact lt_of_le_of_ne (t i).property.1 (fun h => (ht i).1 (Subtype.ext h.symm))
+    · exact lt_of_le_of_ne (t i).property.2 (fun h => (ht i).2 (Subtype.ext h))
+
+omit [Nonempty N] in
+lemma isOpen_interiorCube : IsOpen (interiorCube N) := by
+  have he : interiorCube N = ⋂ i, {t : N → unitInterval | 0 < (t i:ℝ) ∧ (t i:ℝ) < 1} := by
+    ext t; simp [interiorCube]
+  rw [he]
+  apply isOpen_iInter_of_finite
+  intro i
+  exact isOpen_Ioo.preimage (continuous_subtype_val.comp (continuous_apply i))
+
+omit [Fintype N] in
+lemma complement_nonempty : (interiorCube N)ᶜ.Nonempty := by
+  refine ⟨fun _ => 0,?_⟩
+  rw [interiorCube_eq,compl_compl]
+  exact ⟨Classical.choice inferInstance,Or.inl rfl⟩
+
+def interiorHomeomorph : interiorCube N ≃ₜ (N → Ioo (0:ℝ) 1) where
+  toFun t i := ⟨t.1 i,(t.property i)⟩
+  invFun t := ⟨fun i => ⟨(t i:ℝ),(t i).property.1.le,(t i).property.2.le⟩,
+    fun i => (t i).property⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  continuous_toFun := by
+    apply continuous_pi
+    intro i
+    exact (continuous_subtype_val.comp ((continuous_apply i).comp continuous_subtype_val)).subtype_mk _
+  continuous_invFun := by
+    apply Continuous.subtype_mk
+    apply continuous_pi
+    intro i
+    exact (continuous_subtype_val.comp (continuous_apply i)).subtype_mk _
+
+def intervalHomeomorph : Ioo (0:ℝ) 1 ≃ₜ Ioo (-1:ℝ) 1 where
+  toFun x := ⟨2*(x:ℝ)-1,by have h := x.property; constructor <;> linarith [h.1,h.2]⟩
+  invFun y := ⟨((y:ℝ)+1)/2,by have h := y.property; constructor <;> linarith [h.1,h.2]⟩
+  left_inv x := Subtype.ext (by dsimp; ring)
+  right_inv y := Subtype.ext (by dsimp; ring)
+  continuous_toFun := by
+    apply Continuous.subtype_mk
+    exact (continuous_const.mul continuous_subtype_val).sub continuous_const
+  continuous_invFun := by
+    apply Continuous.subtype_mk
+    exact (continuous_subtype_val.add continuous_const).div_const 2
+
+def interiorEuclidean : interiorCube N ≃ₜ (N → ℝ) :=
+  (interiorHomeomorph N).trans (Homeomorph.piCongrRight fun _ =>
+    intervalHomeomorph.trans (orderIsoIooNegOneOne ℝ).toHomeomorph.symm)
+
+def compactificationSphere (n : ℕ) : OnePoint (interiorCube (Fin (n+1))) ≃ₜ
+    Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1 :=
+  (interiorEuclidean (Fin (n+1))).onePointCongr.trans
+    (onePointEquivSphereOfFinrankEq (by simp))
+
+def spherePoint (n : ℕ) : Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1 :=
+  compactificationSphere n ∞
+
+def sphereQuotient (n : ℕ) : C((Fin (n+1) → unitInterval),
+    Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1) :=
+  ⟨fun t => compactificationSphere n (OpenCollapse.collapse (interiorCube _) t),
+    (compactificationSphere n).continuous.comp
+      (OpenCollapse.continuous_collapse (isOpen_interiorCube _))⟩
+
+lemma sphereQuotient_boundary (n : ℕ) {t : Fin (n+1) → unitInterval}
+    (ht : t ∈ Cube.boundary _) : sphereQuotient n t = spherePoint n := by
+  change compactificationSphere n _ = compactificationSphere n ∞
+  rw [OpenCollapse.collapse_of_notMem]
+  simpa only [interiorCube_eq,mem_compl_iff,not_not] using ht
+
+lemma sphereQuotient_surjective (n : ℕ) : Function.Surjective (sphereQuotient n) :=
+  (compactificationSphere n).surjective.comp
+    (OpenCollapse.collapse_surjective (complement_nonempty _))
+
+variable {X : Type*} [TopologicalSpace X] {x : X}
+
+def sphereMap {n : ℕ} (f : GenLoop (Fin (n+1)) X x) :
+    C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X) :=
+  (OpenCollapse.descend (isOpen_interiorCube _) (complement_nonempty _) f.1 x
+      (fun t ht => GenLoop.boundary f t (by simpa only [interiorCube_eq,mem_compl_iff,not_not] using ht))).comp
+    ⟨(compactificationSphere n).symm,(compactificationSphere n).symm.continuous⟩
+
+@[simp] lemma sphereMap_quotient {n : ℕ} (f : GenLoop (Fin (n+1)) X x)
+    (t : Fin (n+1) → unitInterval) : sphereMap f (sphereQuotient n t) = f t := by
+  simp only [sphereMap,ContinuousMap.comp_apply,ContinuousMap.coe_mk,sphereQuotient,
+    Homeomorph.symm_apply_apply,OpenCollapse.descend_collapse]
+  rfl
+
+@[simp] lemma sphereMap_point {n : ℕ} (f : GenLoop (Fin (n+1)) X x) :
+    sphereMap f (spherePoint n) = x := by
+  simp only [sphereMap,ContinuousMap.comp_apply,ContinuousMap.coe_mk,spherePoint,
+    Homeomorph.symm_apply_apply]
+  rfl
+
+theorem sphereMap_null_implies_cube_null {n : ℕ} (f : GenLoop (Fin (n+1)) X x)
+    (hf : (sphereMap f).Nullhomotopic) : GenLoop.Homotopic f GenLoop.const := by
+  obtain ⟨y,⟨H⟩⟩ := hf
+  let H' : f.1.Homotopy (ContinuousMap.const _ y) := {
+    toFun := fun q => H (q.1,sphereQuotient n q.2)
+    continuous_toFun := H.continuous.comp (continuous_fst.prodMk ((sphereQuotient n).continuous.comp continuous_snd))
+    map_zero_left := fun t => (H.apply_zero _).trans (sphereMap_quotient f t)
+    map_one_left := fun t => H.apply_one _ }
+  let p : C(unitInterval,X) :=
+    ⟨fun s => H (s,spherePoint n),H.continuous.comp (continuous_id.prodMk continuous_const)⟩
+  apply CubicalFreeNullity.null_of_moving_base f H' p
+  intro s t ht
+  change H (s,sphereQuotient n t) = H (s,spherePoint n)
+  rw [sphereQuotient_boundary n ht]
+
+theorem sphereMap_null_implies_class_one {n : ℕ} (f : GenLoop (Fin (n+1)) X x)
+    (hf : (sphereMap f).Nullhomotopic) :
+    (⟦f⟧ : HomotopyGroup.Pi (n+1) X x) = (1 : HomotopyGroup.Pi (n+1) X x) := by
+  rw [HomotopyGroup.one_def]
+  exact Quotient.sound (sphereMap_null_implies_cube_null f hf)
+
+end NearbyLagrangian.CubicalSphere
+
+end
+
+noncomputable section
+universe u v w
+open Set ContinuousMap Topology
+open scoped OnePoint Topology Topology.Homotopy
+
+namespace NearbyLagrangian.CubicalSphere
+variable {X : Type*} [TopologicalSpace X] {x : X}
+
+def sphereHomotopy {n : ℕ} {f g : GenLoop (Fin (n+1)) X x}
+    (H : f.1.HomotopyRel g.1 (Cube.boundary _)) : (sphereMap f).Homotopy (sphereMap g) where
+  toFun := fun q => ((compactificationSphere n).symm q.2).elim x
+    (fun t => H (q.1,t))
+  continuous_toFun := by
+    let F : C(unitInterval × (Fin (n+1) → unitInterval),X) := ⟨H,H.continuous⟩
+    have hF : ∀ s t, t ∉ interiorCube (Fin (n+1)) → F (s,t) = x := by
+      intro s t ht
+      have ht' : t ∈ Cube.boundary _ := by
+        simpa only [interiorCube_eq,mem_compl_iff,not_not] using ht
+      exact (H.eq_fst s ht').trans (GenLoop.boundary f t ht')
+    exact (OpenCollapse.continuous_descend_param (isOpen_interiorCube _) (complement_nonempty _)
+      F (const _ x) hF).comp (continuous_fst.prodMk
+        ((compactificationSphere n).symm.continuous.comp continuous_snd))
+  map_zero_left := by
+    intro u
+    change ((compactificationSphere n).symm u).elim x (fun t => H (0,t)) =
+      ((compactificationSphere n).symm u).elim x (fun t => f t)
+    generalize (compactificationSphere n).symm u = z
+    induction z using OnePoint.rec with
+    | infty => rfl
+    | coe t => exact H.apply_zero t
+  map_one_left := by
+    intro u
+    change ((compactificationSphere n).symm u).elim x (fun t => H (1,t)) =
+      ((compactificationSphere n).symm u).elim x (fun t => g t)
+    generalize (compactificationSphere n).symm u = z
+    induction z using OnePoint.rec with
+    | infty => rfl
+    | coe t => exact H.apply_one t
+
+@[simp] theorem sphereMap_const (n : ℕ) : sphereMap (GenLoop.const : GenLoop (Fin (n+1)) X x) =
+    ContinuousMap.const _ x := by
+  ext u
+  obtain ⟨t,rfl⟩ := sphereQuotient_surjective n u
+  exact sphereMap_quotient GenLoop.const t
+
+theorem sphereMap_null_iff_cube_null {n : ℕ} (f : GenLoop (Fin (n+1)) X x) :
+    (sphereMap f).Nullhomotopic ↔ GenLoop.Homotopic f GenLoop.const := by
+  constructor
+  · exact sphereMap_null_implies_cube_null f
+  · rintro ⟨H⟩
+    refine ⟨x,?_⟩
+    rw [← sphereMap_const]
+    exact ⟨sphereHomotopy H⟩
+
+theorem sphereMap_null_iff_class_one {n : ℕ} (f : GenLoop (Fin (n+1)) X x) :
+    (sphereMap f).Nullhomotopic ↔
+      (⟦f⟧ : HomotopyGroup.Pi (n+1) X x) = (1 : HomotopyGroup.Pi (n+1) X x) := by
+  rw [sphereMap_null_iff_cube_null,HomotopyGroup.one_def,Quotient.eq]
+  rfl
+
+end NearbyLagrangian.CubicalSphere
+
+end
+
+noncomputable section
+universe u v w
+
+
+namespace NearbyLagrangian.SphereConnectivity
+open scoped Topology Topology.Homotopy
+
+def genLoopMap {N A B : Type*} [TopologicalSpace A] [TopologicalSpace B]
+    (e : C(A,B)) {x : A} (f : GenLoop N A x) : GenLoop N B (e x) :=
+  ⟨e.comp f.1,fun t ht => congrArg e (GenLoop.boundary f t ht)⟩
+lemma genLoopMap_homotopic {N A B : Type*} [TopologicalSpace A] [TopologicalSpace B]
+    (e : C(A,B)) {x : A} {f g : GenLoop N A x} (h : GenLoop.Homotopic f g) :
+    GenLoop.Homotopic (genLoopMap e f) (genLoopMap e g) := by
+  obtain ⟨H⟩ := h
+  exact ⟨H.compContinuousMap e⟩
+def homotopyGroupMap {n : ℕ} {A B : Type*} [TopologicalSpace A] [TopologicalSpace B]
+    (e : C(A,B)) (x : A) : HomotopyGroup.Pi n A x → HomotopyGroup.Pi n B (e x) :=
+  Quotient.map (genLoopMap e) (fun _ _ h => genLoopMap_homotopic e h)
+end NearbyLagrangian.SphereConnectivity
+
+end
+
+noncomputable section
+universe u v w
+
+
+namespace NearbyLagrangian.CubicalSphere
+open Set ContinuousMap SphereConnectivity
+open scoped Topology Topology.Homotopy
+variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {x : X}
+
+def sphereLoop {n : ℕ}
+    (a : C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X)) :
+    GenLoop (Fin (n+1)) X (a (spherePoint n)) :=
+  ⟨a.comp (sphereQuotient n),fun _t ht => congrArg a (sphereQuotient_boundary n ht)⟩
+
+@[simp] theorem sphereMap_sphereLoop {n : ℕ}
+    (a : C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X)) :
+    sphereMap (sphereLoop a) = a := by
+  ext u
+  obtain ⟨t,rfl⟩ := sphereQuotient_surjective n u
+  rw [sphereMap_quotient]
+  rfl
+
+@[simp] theorem sphereMap_natural {n : ℕ} (e : C(X,Y))
+    (f : GenLoop (Fin (n+1)) X x) :
+    sphereMap (genLoopMap e f) = e.comp (sphereMap f) := by
+  ext u
+  obtain ⟨t,rfl⟩ := sphereQuotient_surjective n u
+  rw [sphereMap_quotient,ContinuousMap.comp_apply,sphereMap_quotient]
+  rfl
+
+theorem null_iff_class_one {n : ℕ}
+    (a : C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X)) :
+    a.Nullhomotopic ↔
+      (⟦sphereLoop a⟧ : HomotopyGroup.Pi (n+1) X (a (spherePoint n))) =
+        (1 : HomotopyGroup.Pi (n+1) X (a (spherePoint n))) := by
+  simpa only [sphereMap_sphereLoop] using sphereMap_null_iff_class_one (sphereLoop a)
+
+theorem comp_null_iff_mapped_class_one {n : ℕ} (e : C(X,Y))
+    (a : C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X)) :
+    (e.comp a).Nullhomotopic ↔
+      homotopyGroupMap (n:=n+1) e (a (spherePoint n)) ⟦sphereLoop a⟧ =
+        (1 : HomotopyGroup.Pi (n+1) Y (e (a (spherePoint n)))) := by
+  change (e.comp a).Nullhomotopic ↔
+    (⟦genLoopMap e (sphereLoop a)⟧ : HomotopyGroup.Pi (n+1) Y (e (a (spherePoint n)))) = _
+  simpa only [sphereMap_natural,sphereMap_sphereLoop] using
+    sphereMap_null_iff_class_one (genLoopMap e (sphereLoop a))
+
+end NearbyLagrangian.CubicalSphere
+
+end
+
+noncomputable section
+universe u v w
+open scoped Classical Topology Topology.Homotopy
+
+namespace NearbyLagrangian.CubicalSphere
+open Set ContinuousMap
+open scoped OnePoint
+variable {X : Type*} [TopologicalSpace X] {x y : X}
+
+def sphereMovingHomotopy {n : ℕ} {f : GenLoop (Fin (n+1)) X x}
+    {g : GenLoop (Fin (n+1)) X y} (H : f.1.Homotopy g.1)
+    (p : Path x y) (hp : ∀ s t, t ∈ Cube.boundary (Fin (n+1)) → H (s,t) = p s) :
+    (sphereMap f).Homotopy (sphereMap g) where
+  toFun q := ((compactificationSphere n).symm q.2).elim (p q.1) (fun t => H (q.1,t))
+  continuous_toFun := by
+    let F : C(unitInterval × (Fin (n+1) → unitInterval),X) := ⟨H,H.continuous⟩
+    have hF (s : unitInterval) (t : Fin (n+1) → unitInterval)
+        (ht : t ∉ interiorCube (Fin (n+1))) : F (s,t) = p s := by
+      apply hp
+      simpa only [interiorCube_eq,mem_compl_iff,not_not] using ht
+    exact (OpenCollapse.continuous_descend_param (isOpen_interiorCube _) (complement_nonempty _)
+      F p.toContinuousMap hF).comp (continuous_fst.prodMk
+        ((compactificationSphere n).symm.continuous.comp continuous_snd))
+  map_zero_left := by
+    intro u
+    change ((compactificationSphere n).symm u).elim (p 0) (fun t => H (0,t)) =
+      ((compactificationSphere n).symm u).elim x (fun t => f t)
+    rw [p.source]
+    generalize (compactificationSphere n).symm u = z
+    induction z using OnePoint.rec with
+    | infty => rfl
+    | coe t => exact H.apply_zero t
+  map_one_left := by
+    intro u
+    change ((compactificationSphere n).symm u).elim (p 1) (fun t => H (1,t)) =
+      ((compactificationSphere n).symm u).elim y (fun t => g t)
+    rw [p.target]
+    generalize (compactificationSphere n).symm u = z
+    induction z using OnePoint.rec with
+    | infty => rfl
+    | coe t => exact H.apply_one t
+
+theorem exists_based_representative [PathConnectedSpace X] {n : ℕ}
+    (a : C(Metric.sphere (0 : EuclideanSpace ℝ (Fin (n+2))) 1,X)) (x : X) :
+    ∃ f : GenLoop (Fin (n+1)) X x, a.Homotopic (sphereMap f) := by
+  obtain ⟨p⟩ := PathConnectedSpace.joined (a (spherePoint n)) x
+  refine ⟨CubicalFreeNullity.rebase (sphereLoop a) p,?_⟩
+  have h := sphereMovingHomotopy (CubicalFreeNullity.rebaseHomotopy (sphereLoop a) p) p
+    (fun s _t ht => CubicalFreeNullity.rebaseValue_boundary (sphereLoop a) p s ht)
+  simpa only [sphereMap_sphereLoop] using (show (sphereMap (sphereLoop a)).Homotopic
+    (sphereMap (CubicalFreeNullity.rebase (sphereLoop a) p)) from ⟨h⟩)
+
+end NearbyLagrangian.CubicalSphere
+
+end
+
+end OAI

lean/OAI/Topology/Homotopy/FiberCone.lean

@@ -1,393 +1,393 @@
-import Mathlib
-
-namespace OAI
-
-/-! Fiberwise cones, contractions and left suspension homotopies. -/
-
-noncomputable section
-universe u v w
-open Set ContinuousMap
-open scoped Topology unitInterval
-
-namespace NearbyLagrangian.FiberCone
-
-variable {B A X Y : Type*} [TopologicalSpace B] [TopologicalSpace A]
-  [TopologicalSpace X] [TopologicalSpace Y]
-
-abbrev Raw (B A X : Type*) := X ⊕ ((unitInterval × A) ⊕ B)
-
-inductive Seam (f : C(A,X)) (q : C(A,B)) : Raw B A X → Raw B A X → Prop
-  | bottom (a : A) : Seam f q (.inr (.inl (0,a))) (.inl (f a))
-  | top (a : A) : Seam f q (.inr (.inl (1,a))) (.inr (.inr (q a)))
-
-def Space (f : C(A,X)) (q : C(A,B)) := Quot (Seam f q)
-instance (f : C(A,X)) (q : C(A,B)) : TopologicalSpace (Space f q) :=
-  inferInstanceAs (TopologicalSpace (Quot (Seam f q)))
-
-def base (f : C(A,X)) (q : C(A,B)) : C(X,Space f q) :=
-  ⟨fun x => Quot.mk _ (.inl x),continuous_quot_mk.comp continuous_inl⟩
-def ray (f : C(A,X)) (q : C(A,B)) : C(unitInterval × A,Space f q) :=
-  ⟨fun z => Quot.mk _ (.inr (.inl z)),
-    continuous_quot_mk.comp (continuous_inr.comp continuous_inl)⟩
-def apex (f : C(A,X)) (q : C(A,B)) : C(B,Space f q) :=
-  ⟨fun b => Quot.mk _ (.inr (.inr b)),
-    continuous_quot_mk.comp (continuous_inr.comp continuous_inr)⟩
-
-@[simp] lemma ray_zero (f : C(A,X)) (q : C(A,B)) (a : A) :
-    ray f q (0,a) = base f q (f a) := Quot.sound (Seam.bottom a)
-@[simp] lemma ray_one (f : C(A,X)) (q : C(A,B)) (a : A) :
-    ray f q (1,a) = apex f q (q a) := Quot.sound (Seam.top a)
-
-def lift (f : C(A,X)) (q : C(A,B)) (i : C(X,Y))
-    (c : C(unitInterval × A,Y)) (v : C(B,Y))
-    (h₀ : ∀ a, c (0,a) = i (f a)) (h₁ : ∀ a, c (1,a) = v (q a)) :
-    C(Space f q,Y) where
-  toFun := Quot.lift (Sum.elim i (Sum.elim c v)) (by
-    intro a b h
-    cases h with
-    | bottom a => exact h₀ a
-    | top a => exact h₁ a)
-  continuous_toFun := continuous_quot_lift _
-    (i.continuous.sumElim (c.continuous.sumElim v.continuous))
-
-@[simp] lemma lift_base (f : C(A,X)) (q : C(A,B)) (i : C(X,Y))
-    (c : C(unitInterval × A,Y)) (v : C(B,Y))
-    (h₀ : ∀ a, c (0,a) = i (f a)) (h₁ : ∀ a, c (1,a) = v (q a)) (x : X) :
-    lift f q i c v h₀ h₁ (base f q x) = i x := rfl
-@[simp] lemma lift_ray (f : C(A,X)) (q : C(A,B)) (i : C(X,Y))
-    (c : C(unitInterval × A,Y)) (v : C(B,Y))
-    (h₀ : ∀ a, c (0,a) = i (f a)) (h₁ : ∀ a, c (1,a) = v (q a)) (z : unitInterval × A) :
-    lift f q i c v h₀ h₁ (ray f q z) = c z := rfl
-@[simp] lemma lift_apex (f : C(A,X)) (q : C(A,B)) (i : C(X,Y))
-    (c : C(unitInterval × A,Y)) (v : C(B,Y))
-    (h₀ : ∀ a, c (0,a) = i (f a)) (h₁ : ∀ a, c (1,a) = v (q a)) (b : B) :
-    lift f q i c v h₀ h₁ (apex f q b) = v b := rfl
-
-@[elab_as_elim] theorem induction {f : C(A,X)} {q : C(A,B)}
-    {P : Space f q → Prop} (hbase : ∀ x, P (base f q x))
-    (hray : ∀ ta, P (ray f q ta)) (hapex : ∀ b, P (apex f q b)) (z : Space f q) : P z := by
-  induction z using Quot.inductionOn with
-  | h z =>
-    rcases z with x | (ta | b)
-    · exact hbase x
-    · exact hray ta
-    · exact hapex b
-
-def projection (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) : C(Space f q,B) :=
-  lift f q p (q.comp ⟨Prod.snd,continuous_snd⟩) (ContinuousMap.id B)
-    (fun a => (hf a).symm) (fun _ => rfl)
-
-@[simp] lemma projection_base (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) (x : X) : projection f p q hf (base f q x) = p x := rfl
-@[simp] lemma projection_ray (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) (ta : unitInterval × A) :
-    projection f p q hf (ray f q ta) = q ta.2 := rfl
-@[simp] lemma projection_apex (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) (b : B) : projection f p q hf (apex f q b) = b := rfl
-
-abbrev Susp (q : C(A,B)) := Space q q
-
-def suspProjection (q : C(A,B)) : C(Susp q,B) :=
-  projection q (ContinuousMap.id B) q (fun _ => rfl)
-
-abbrev south (q : C(A,B)) : C(B,Susp q) := base q q
-abbrev meridian (q : C(A,B)) : C(unitInterval × A,Susp q) := ray q q
-abbrev north (q : C(A,B)) : C(B,Susp q) := apex q q
-
-def suspMap (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) : C(Susp q,Susp p) :=
-  lift q q (south p)
-    ((meridian p).comp ⟨fun ta => (ta.1,f ta.2),continuous_fst.prodMk
-      (f.continuous.comp continuous_snd)⟩) (north p)
-    (fun a => by change meridian p (0,f a) = south p (q a); rw [ray_zero,hf])
-    (fun a => by change meridian p (1,f a) = north p (q a); rw [ray_one,hf])
-
-@[simp] lemma suspMap_south (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) (b : B) : suspMap f p q hf (south q b) = south p b := rfl
-@[simp] lemma suspMap_meridian (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) (ta : unitInterval × A) :
-    suspMap f p q hf (meridian q ta) = meridian p (ta.1,f ta.2) := rfl
-@[simp] lemma suspMap_north (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) (b : B) : suspMap f p q hf (north q b) = north p b := rfl
-
-lemma suspMap_over (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) (z : Susp q) :
-    suspProjection p (suspMap f p q hf z) = suspProjection q z := by
-  induction z using induction with
-  | hbase b => rfl
-  | hray ta => exact hf ta.2
-  | hapex b => rfl
-
-def collapse (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) : C(Space f q,Susp q) :=
-  lift f q ((south q).comp p) (meridian q) (north q)
-    (fun a => by change meridian q (0,a) = south q (p (f a)); rw [ray_zero,hf])
-    (fun a => ray_one q q a)
-
-@[simp] lemma collapse_base (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) (x : X) : collapse f p q hf (base f q x) = south q (p x) := rfl
-@[simp] lemma collapse_ray (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) (ta : unitInterval × A) : collapse f p q hf (ray f q ta) = meridian q ta := rfl
-@[simp] lemma collapse_apex (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) (b : B) : collapse f p q hf (apex f q b) = north q b := rfl
-
-lemma collapse_over (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) (z : Space f q) :
-    suspProjection q (collapse f p q hf z) = projection f p q hf z := by
-  induction z using induction with
-  | hbase x => rfl
-  | hray ta => rfl
-  | hapex b => rfl
-
-end NearbyLagrangian.FiberCone
-
-end
-
-noncomputable section
-universe u v w
-open Set ContinuousMap
-open scoped Topology unitInterval
-
-namespace NearbyLagrangian.FiberCone
-variable {B A X : Type*} [TopologicalSpace B] [TopologicalSpace A] [TopologicalSpace X]
-
-structure Contraction (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
-    (hf : ∀ a, p (f a) = q a) where
-  hom : C(unitInterval × Space f q,Space f q)
-  at_zero : ∀ z, hom (0,z) = z
-  at_one : ∀ z, hom (1,z) = apex f q (projection f p q hf z)
-  staysOver : ∀ t z, projection f p q hf (hom (t,z)) = projection f p q hf z
-  fixed : ∀ t b, hom (t,apex f q b) = apex f q b
-
-namespace Contraction
-variable {f : C(A,X)} {p : C(X,B)} {q : C(A,B)} {hf : ∀ a, p (f a) = q a}
-variable (H : Contraction f p q hf)
-
-def inverseRay : C(unitInterval × X,Susp q) :=
-  (collapse f p q hf).comp
-    (H.hom.comp ⟨fun tx => (tx.1,base f q tx.2),
-      continuous_fst.prodMk ((base f q).continuous.comp continuous_snd)⟩)
-
-@[simp] lemma inverseRay_zero (x : X) : H.inverseRay (0,x) = south q (p x) := by
-  change collapse f p q hf (H.hom (0,base f q x)) = _
-  rw [H.at_zero,collapse_base]
-@[simp] lemma inverseRay_one (x : X) : H.inverseRay (1,x) = north q (p x) := by
-  change collapse f p q hf (H.hom (1,base f q x)) = _
-  rw [H.at_one,projection_base,collapse_apex]
-
-def inverse : C(Susp p,Susp q) :=
-  lift p p (south q) H.inverseRay (north q) H.inverseRay_zero H.inverseRay_one
-
-@[simp] lemma inverse_south (b : B) : H.inverse (south p b) = south q b := rfl
-@[simp] lemma inverse_meridian (tx : unitInterval × X) :
-    H.inverse (meridian p tx) = collapse f p q hf (H.hom (tx.1,base f q tx.2)) := rfl
-@[simp] lemma inverse_north (b : B) : H.inverse (north p b) = north q b := rfl
-
-lemma inverse_over (z : Susp p) : suspProjection q (H.inverse z) = suspProjection p z := by
-  induction z using induction with
-  | hbase b => rfl
-  | hray tx =>
-    rw [inverse_meridian,collapse_over,H.staysOver,projection_base]
-    rfl
-  | hapex b => rfl
-
-end Contraction
-
-def squareDen (s : unitInterval) : ℝ := max (s:ℝ) (1-(s:ℝ))
-lemma squareDen_pos (s : unitInterval) : 0 < squareDen s := by
-  have ha := le_max_left (s:ℝ) (1-(s:ℝ))
-  have hb := le_max_right (s:ℝ) (1-(s:ℝ))
-  dsimp [squareDen]
-  linarith
-lemma continuous_squareDen : Continuous squareDen :=
-  continuous_subtype_val.max (continuous_const.sub continuous_subtype_val)
-
-def horizontal : C(unitInterval × unitInterval,unitInterval) where
-  toFun st := ⟨(1-(st.1:ℝ))*(st.2:ℝ) / squareDen st.1,by
-    refine ⟨div_nonneg (mul_nonneg (sub_nonneg.mpr st.1.property.2) st.2.property.1)
-      (squareDen_pos st.1).le,?_⟩
-    apply (div_le_one (squareDen_pos st.1)).mpr
-    have h := mul_le_mul_of_nonneg_left st.2.property.2 (sub_nonneg.mpr st.1.property.2)
-    have h' : (1 - (st.1:ℝ)) * (st.2:ℝ) ≤ 1 - (st.1:ℝ) := by
-      simpa only [mul_one] using h
-    exact h'.trans (le_max_right _ _)⟩
-  continuous_toFun := by
-    apply Continuous.subtype_mk
-    exact (((continuous_const.sub (continuous_subtype_val.comp continuous_fst)).mul
-    (continuous_subtype_val.comp continuous_snd)).div
-      (continuous_squareDen.comp continuous_fst) (fun _ => (squareDen_pos _).ne'))
-
-def vertical : C(unitInterval × unitInterval,unitInterval) where
-  toFun st := ⟨(st.1:ℝ)*(st.2:ℝ) / squareDen st.1,by
-    refine ⟨div_nonneg (mul_nonneg st.1.property.1 st.2.property.1) (squareDen_pos st.1).le,?_⟩
-    apply (div_le_one (squareDen_pos st.1)).mpr
-    have h := mul_le_mul_of_nonneg_left st.2.property.2 st.1.property.1
-    have h' : (st.1:ℝ) * (st.2:ℝ) ≤ (st.1:ℝ) := by
-      simpa only [mul_one] using h
-    exact h'.trans (le_max_left _ _)⟩
-  continuous_toFun := by
-    apply Continuous.subtype_mk
-    exact (((continuous_subtype_val.comp continuous_fst).mul
-    (continuous_subtype_val.comp continuous_snd)).div
-      (continuous_squareDen.comp continuous_fst) (fun _ => (squareDen_pos _).ne'))
-
-@[simp] lemma horizontal_left (t : unitInterval) : horizontal (0,t) = t := by
-  apply Subtype.ext
-  simp [horizontal,squareDen]
-@[simp] lemma horizontal_right (t : unitInterval) : horizontal (1,t) = 0 := by
-  apply Subtype.ext
-  simp [horizontal,squareDen]
-@[simp] lemma vertical_left (t : unitInterval) : vertical (0,t) = 0 := by
-  apply Subtype.ext
-  simp [vertical,squareDen]
-@[simp] lemma vertical_right (t : unitInterval) : vertical (1,t) = t := by
-  apply Subtype.ext
-  simp [vertical,squareDen]
-@[simp] lemma horizontal_bottom (s : unitInterval) : horizontal (s,0) = 0 := by
-  apply Subtype.ext
-  simp [horizontal]
-@[simp] lemma vertical_bottom (s : unitInterval) : vertical (s,0) = 0 := by
-  apply Subtype.ext
-  simp [vertical]
-
-lemma square_top (s : unitInterval) : horizontal (s,1) = 1 ∨ vertical (s,1) = 1 := by
-  rcases le_total (s:ℝ) (1-(s:ℝ)) with hs | hs
-  · left
-    apply Subtype.ext
-    have hn : 1-(s:ℝ) ≠ 0 := by
-      have hd := squareDen_pos s
-      rw [squareDen,max_eq_right hs] at hd
-      exact hd.ne'
-    simp [horizontal,squareDen,max_eq_right hs,hn]
-  · right
-    apply Subtype.ext
-    have hn : (s:ℝ) ≠ 0 := by
-      have hd := squareDen_pos s
-      rw [squareDen,max_eq_left hs] at hd
-      exact hd.ne'
-    simp [vertical,squareDen,max_eq_left hs,hn]
-
-end NearbyLagrangian.FiberCone
-
-end
-
-noncomputable section
-universe u v w
-open Set ContinuousMap
-open scoped Topology unitInterval
-
-namespace NearbyLagrangian.FiberCone
-variable {B A X Y : Type*} [TopologicalSpace B] [TopologicalSpace A]
-  [TopologicalSpace X] [TopologicalSpace Y]
-
-def cylinderLift (f : C(A,X)) (q : C(A,B)) (i : C(unitInterval × X,Y))
-    (c : C(unitInterval × (unitInterval × A),Y)) (v : C(unitInterval × B,Y))
-    (h₀ : ∀ s a, c (s,(0,a)) = i (s,f a))
-    (h₁ : ∀ s a, c (s,(1,a)) = v (s,q a)) : C(unitInterval × Space f q,Y) :=
-  (ContinuousMap.uncurry (lift f q
-    (ContinuousMap.curry (i.comp ContinuousMap.prodSwap))
-    (ContinuousMap.curry (c.comp ContinuousMap.prodSwap))
-    (ContinuousMap.curry (v.comp ContinuousMap.prodSwap))
-    (fun a => by ext s; exact h₀ s a)
-    (fun a => by ext s; exact h₁ s a))).comp ContinuousMap.prodSwap
-
-@[simp] lemma cylinderLift_base (f : C(A,X)) (q : C(A,B)) (i : C(unitInterval × X,Y))
-    (c : C(unitInterval × (unitInterval × A),Y)) (v : C(unitInterval × B,Y))
-    (h₀ : ∀ s a, c (s,(0,a)) = i (s,f a))
-    (h₁ : ∀ s a, c (s,(1,a)) = v (s,q a)) (s : unitInterval) (x : X) :
-    cylinderLift f q i c v h₀ h₁ (s,base f q x) = i (s,x) := rfl
-@[simp] lemma cylinderLift_ray (f : C(A,X)) (q : C(A,B)) (i : C(unitInterval × X,Y))
-    (c : C(unitInterval × (unitInterval × A),Y)) (v : C(unitInterval × B,Y))
-    (h₀ : ∀ s a, c (s,(0,a)) = i (s,f a))
-    (h₁ : ∀ s a, c (s,(1,a)) = v (s,q a)) (s : unitInterval) (ta : unitInterval × A) :
-    cylinderLift f q i c v h₀ h₁ (s,ray f q ta) = c (s,ta) := rfl
-@[simp] lemma cylinderLift_apex (f : C(A,X)) (q : C(A,B)) (i : C(unitInterval × X,Y))
-    (c : C(unitInterval × (unitInterval × A),Y)) (v : C(unitInterval × B,Y))
-    (h₀ : ∀ s a, c (s,(0,a)) = i (s,f a))
-    (h₁ : ∀ s a, c (s,(1,a)) = v (s,q a)) (s : unitInterval) (b : B) :
-    cylinderLift f q i c v h₀ h₁ (s,apex f q b) = v (s,b) := rfl
-
-namespace Contraction
-variable {f : C(A,X)} {p : C(X,B)} {q : C(A,B)} {hf : ∀ a, p (f a) = q a}
-variable (H : Contraction f p q hf)
-
-def leftRay : C(unitInterval × (unitInterval × A),Susp q) where
-  toFun z := collapse f p q hf (H.hom
-    (vertical (z.1,z.2.1),ray f q (horizontal (z.1,z.2.1),z.2.2)))
-  continuous_toFun := (collapse f p q hf).continuous.comp (H.hom.continuous.comp
-    ((vertical.continuous.comp (continuous_fst.prodMk (continuous_fst.comp continuous_snd))).prodMk
-      ((ray f q).continuous.comp
-        ((horizontal.continuous.comp (continuous_fst.prodMk (continuous_fst.comp continuous_snd))).prodMk
-          (continuous_snd.comp continuous_snd)))))
-
-lemma leftRay_bottom (s : unitInterval) (a : A) : H.leftRay (s,(0,a)) = south q (q a) := by
-  change collapse f p q hf (H.hom (vertical (s,0),ray f q (horizontal (s,0),a))) = _
-  rw [horizontal_bottom,vertical_bottom,H.at_zero,ray_zero,collapse_base,hf]
-
-lemma leftRay_top (s : unitInterval) (a : A) : H.leftRay (s,(1,a)) = north q (q a) := by
-  change collapse f p q hf (H.hom (vertical (s,1),ray f q (horizontal (s,1),a))) = _
-  rcases square_top s with hx | hy
-  · rw [hx,ray_one,H.fixed,collapse_apex]
-  · rw [hy,H.at_one,projection_ray,collapse_apex]
-
-def leftDeformation : C(unitInterval × Susp q,Susp q) :=
-  cylinderLift q q ((south q).comp ⟨Prod.snd,continuous_snd⟩) H.leftRay
-    ((north q).comp ⟨Prod.snd,continuous_snd⟩) H.leftRay_bottom H.leftRay_top
-
-@[simp] lemma leftDeformation_south (s : unitInterval) (b : B) :
-    H.leftDeformation (s,south q b) = south q b := rfl
-@[simp] lemma leftDeformation_meridian (s : unitInterval) (ta : unitInterval × A) :
-    H.leftDeformation (s,meridian q ta) = H.leftRay (s,ta) := rfl
-@[simp] lemma leftDeformation_north (s : unitInterval) (b : B) :
-    H.leftDeformation (s,north q b) = north q b := rfl
-
-lemma leftDeformation_zero (z : Susp q) : H.leftDeformation (0,z) = z := by
-  induction z using induction with
-  | hbase b => rfl
-  | hray ta =>
-    rw [leftDeformation_meridian]
-    change collapse f p q hf (H.hom
-      (vertical (0,ta.1),ray f q (horizontal (0,ta.1),ta.2))) = _
-    rw [horizontal_left,vertical_left,H.at_zero,collapse_ray]
-  | hapex b => rfl
-
-lemma leftDeformation_one (z : Susp q) :
-    H.leftDeformation (1,z) = H.inverse (suspMap f p q hf z) := by
-  induction z using induction with
-  | hbase b => rw [leftDeformation_south,suspMap_south,inverse_south]
-  | hray ta =>
-    rw [leftDeformation_meridian,suspMap_meridian,inverse_meridian]
-    change collapse f p q hf (H.hom
-      (vertical (1,ta.1),ray f q (horizontal (1,ta.1),ta.2))) = _
-    rw [horizontal_right,vertical_right,ray_zero]
-  | hapex b => rw [leftDeformation_north,suspMap_north,inverse_north]
-
-lemma leftDeformation_over (s : unitInterval) (z : Susp q) :
-    suspProjection q (H.leftDeformation (s,z)) = suspProjection q z := by
-  induction z using induction with
-  | hbase b => rfl
-  | hray ta =>
-    rw [leftDeformation_meridian]
-    change suspProjection q (collapse f p q hf (H.hom
-      (vertical (s,ta.1),ray f q (horizontal (s,ta.1),ta.2)))) = _
-    rw [collapse_over,H.staysOver,projection_ray]
-    rfl
-  | hapex b => rfl
-
-def leftHomotopy : (H.inverse.comp (suspMap f p q hf)).Homotopy (ContinuousMap.id (Susp q)) :=
-  ContinuousMap.Homotopy.symm
-    { toContinuousMap := H.leftDeformation
-      map_zero_left := H.leftDeformation_zero
-      map_one_left := H.leftDeformation_one }
-
-lemma leftHomotopy_over (s : unitInterval) (z : Susp q) :
-    suspProjection q (H.leftHomotopy (s,z)) = suspProjection q z :=
-  H.leftDeformation_over (unitInterval.symm s) z
-
-end Contraction
-end NearbyLagrangian.FiberCone
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Fiberwise cones, contractions and left suspension homotopies. -/
+
+noncomputable section
+universe u v w
+open Set ContinuousMap
+open scoped Topology unitInterval
+
+namespace NearbyLagrangian.FiberCone
+
+variable {B A X Y : Type*} [TopologicalSpace B] [TopologicalSpace A]
+  [TopologicalSpace X] [TopologicalSpace Y]
+
+abbrev Raw (B A X : Type*) := X ⊕ ((unitInterval × A) ⊕ B)
+
+inductive Seam (f : C(A,X)) (q : C(A,B)) : Raw B A X → Raw B A X → Prop
+  | bottom (a : A) : Seam f q (.inr (.inl (0,a))) (.inl (f a))
+  | top (a : A) : Seam f q (.inr (.inl (1,a))) (.inr (.inr (q a)))
+
+def Space (f : C(A,X)) (q : C(A,B)) := Quot (Seam f q)
+instance (f : C(A,X)) (q : C(A,B)) : TopologicalSpace (Space f q) :=
+  inferInstanceAs (TopologicalSpace (Quot (Seam f q)))
+
+def base (f : C(A,X)) (q : C(A,B)) : C(X,Space f q) :=
+  ⟨fun x => Quot.mk _ (.inl x),continuous_quot_mk.comp continuous_inl⟩
+def ray (f : C(A,X)) (q : C(A,B)) : C(unitInterval × A,Space f q) :=
+  ⟨fun z => Quot.mk _ (.inr (.inl z)),
+    continuous_quot_mk.comp (continuous_inr.comp continuous_inl)⟩
+def apex (f : C(A,X)) (q : C(A,B)) : C(B,Space f q) :=
+  ⟨fun b => Quot.mk _ (.inr (.inr b)),
+    continuous_quot_mk.comp (continuous_inr.comp continuous_inr)⟩
+
+@[simp] lemma ray_zero (f : C(A,X)) (q : C(A,B)) (a : A) :
+    ray f q (0,a) = base f q (f a) := Quot.sound (Seam.bottom a)
+@[simp] lemma ray_one (f : C(A,X)) (q : C(A,B)) (a : A) :
+    ray f q (1,a) = apex f q (q a) := Quot.sound (Seam.top a)
+
+def lift (f : C(A,X)) (q : C(A,B)) (i : C(X,Y))
+    (c : C(unitInterval × A,Y)) (v : C(B,Y))
+    (h₀ : ∀ a, c (0,a) = i (f a)) (h₁ : ∀ a, c (1,a) = v (q a)) :
+    C(Space f q,Y) where
+  toFun := Quot.lift (Sum.elim i (Sum.elim c v)) (by
+    intro a b h
+    cases h with
+    | bottom a => exact h₀ a
+    | top a => exact h₁ a)
+  continuous_toFun := continuous_quot_lift _
+    (i.continuous.sumElim (c.continuous.sumElim v.continuous))
+
+@[simp] lemma lift_base (f : C(A,X)) (q : C(A,B)) (i : C(X,Y))
+    (c : C(unitInterval × A,Y)) (v : C(B,Y))
+    (h₀ : ∀ a, c (0,a) = i (f a)) (h₁ : ∀ a, c (1,a) = v (q a)) (x : X) :
+    lift f q i c v h₀ h₁ (base f q x) = i x := rfl
+@[simp] lemma lift_ray (f : C(A,X)) (q : C(A,B)) (i : C(X,Y))
+    (c : C(unitInterval × A,Y)) (v : C(B,Y))
+    (h₀ : ∀ a, c (0,a) = i (f a)) (h₁ : ∀ a, c (1,a) = v (q a)) (z : unitInterval × A) :
+    lift f q i c v h₀ h₁ (ray f q z) = c z := rfl
+@[simp] lemma lift_apex (f : C(A,X)) (q : C(A,B)) (i : C(X,Y))
+    (c : C(unitInterval × A,Y)) (v : C(B,Y))
+    (h₀ : ∀ a, c (0,a) = i (f a)) (h₁ : ∀ a, c (1,a) = v (q a)) (b : B) :
+    lift f q i c v h₀ h₁ (apex f q b) = v b := rfl
+
+@[elab_as_elim] theorem induction {f : C(A,X)} {q : C(A,B)}
+    {P : Space f q → Prop} (hbase : ∀ x, P (base f q x))
+    (hray : ∀ ta, P (ray f q ta)) (hapex : ∀ b, P (apex f q b)) (z : Space f q) : P z := by
+  induction z using Quot.inductionOn with
+  | h z =>
+    rcases z with x | (ta | b)
+    · exact hbase x
+    · exact hray ta
+    · exact hapex b
+
+def projection (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) : C(Space f q,B) :=
+  lift f q p (q.comp ⟨Prod.snd,continuous_snd⟩) (ContinuousMap.id B)
+    (fun a => (hf a).symm) (fun _ => rfl)
+
+@[simp] lemma projection_base (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) (x : X) : projection f p q hf (base f q x) = p x := rfl
+@[simp] lemma projection_ray (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) (ta : unitInterval × A) :
+    projection f p q hf (ray f q ta) = q ta.2 := rfl
+@[simp] lemma projection_apex (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) (b : B) : projection f p q hf (apex f q b) = b := rfl
+
+abbrev Susp (q : C(A,B)) := Space q q
+
+def suspProjection (q : C(A,B)) : C(Susp q,B) :=
+  projection q (ContinuousMap.id B) q (fun _ => rfl)
+
+abbrev south (q : C(A,B)) : C(B,Susp q) := base q q
+abbrev meridian (q : C(A,B)) : C(unitInterval × A,Susp q) := ray q q
+abbrev north (q : C(A,B)) : C(B,Susp q) := apex q q
+
+def suspMap (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) : C(Susp q,Susp p) :=
+  lift q q (south p)
+    ((meridian p).comp ⟨fun ta => (ta.1,f ta.2),continuous_fst.prodMk
+      (f.continuous.comp continuous_snd)⟩) (north p)
+    (fun a => by change meridian p (0,f a) = south p (q a); rw [ray_zero,hf])
+    (fun a => by change meridian p (1,f a) = north p (q a); rw [ray_one,hf])
+
+@[simp] lemma suspMap_south (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) (b : B) : suspMap f p q hf (south q b) = south p b := rfl
+@[simp] lemma suspMap_meridian (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) (ta : unitInterval × A) :
+    suspMap f p q hf (meridian q ta) = meridian p (ta.1,f ta.2) := rfl
+@[simp] lemma suspMap_north (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) (b : B) : suspMap f p q hf (north q b) = north p b := rfl
+
+lemma suspMap_over (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) (z : Susp q) :
+    suspProjection p (suspMap f p q hf z) = suspProjection q z := by
+  induction z using induction with
+  | hbase b => rfl
+  | hray ta => exact hf ta.2
+  | hapex b => rfl
+
+def collapse (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) : C(Space f q,Susp q) :=
+  lift f q ((south q).comp p) (meridian q) (north q)
+    (fun a => by change meridian q (0,a) = south q (p (f a)); rw [ray_zero,hf])
+    (fun a => ray_one q q a)
+
+@[simp] lemma collapse_base (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) (x : X) : collapse f p q hf (base f q x) = south q (p x) := rfl
+@[simp] lemma collapse_ray (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) (ta : unitInterval × A) : collapse f p q hf (ray f q ta) = meridian q ta := rfl
+@[simp] lemma collapse_apex (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) (b : B) : collapse f p q hf (apex f q b) = north q b := rfl
+
+lemma collapse_over (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) (z : Space f q) :
+    suspProjection q (collapse f p q hf z) = projection f p q hf z := by
+  induction z using induction with
+  | hbase x => rfl
+  | hray ta => rfl
+  | hapex b => rfl
+
+end NearbyLagrangian.FiberCone
+
+end
+
+noncomputable section
+universe u v w
+open Set ContinuousMap
+open scoped Topology unitInterval
+
+namespace NearbyLagrangian.FiberCone
+variable {B A X : Type*} [TopologicalSpace B] [TopologicalSpace A] [TopologicalSpace X]
+
+structure Contraction (f : C(A,X)) (p : C(X,B)) (q : C(A,B))
+    (hf : ∀ a, p (f a) = q a) where
+  hom : C(unitInterval × Space f q,Space f q)
+  at_zero : ∀ z, hom (0,z) = z
+  at_one : ∀ z, hom (1,z) = apex f q (projection f p q hf z)
+  staysOver : ∀ t z, projection f p q hf (hom (t,z)) = projection f p q hf z
+  fixed : ∀ t b, hom (t,apex f q b) = apex f q b
+
+namespace Contraction
+variable {f : C(A,X)} {p : C(X,B)} {q : C(A,B)} {hf : ∀ a, p (f a) = q a}
+variable (H : Contraction f p q hf)
+
+def inverseRay : C(unitInterval × X,Susp q) :=
+  (collapse f p q hf).comp
+    (H.hom.comp ⟨fun tx => (tx.1,base f q tx.2),
+      continuous_fst.prodMk ((base f q).continuous.comp continuous_snd)⟩)
+
+@[simp] lemma inverseRay_zero (x : X) : H.inverseRay (0,x) = south q (p x) := by
+  change collapse f p q hf (H.hom (0,base f q x)) = _
+  rw [H.at_zero,collapse_base]
+@[simp] lemma inverseRay_one (x : X) : H.inverseRay (1,x) = north q (p x) := by
+  change collapse f p q hf (H.hom (1,base f q x)) = _
+  rw [H.at_one,projection_base,collapse_apex]
+
+def inverse : C(Susp p,Susp q) :=
+  lift p p (south q) H.inverseRay (north q) H.inverseRay_zero H.inverseRay_one
+
+@[simp] lemma inverse_south (b : B) : H.inverse (south p b) = south q b := rfl
+@[simp] lemma inverse_meridian (tx : unitInterval × X) :
+    H.inverse (meridian p tx) = collapse f p q hf (H.hom (tx.1,base f q tx.2)) := rfl
+@[simp] lemma inverse_north (b : B) : H.inverse (north p b) = north q b := rfl
+
+lemma inverse_over (z : Susp p) : suspProjection q (H.inverse z) = suspProjection p z := by
+  induction z using induction with
+  | hbase b => rfl
+  | hray tx =>
+    rw [inverse_meridian,collapse_over,H.staysOver,projection_base]
+    rfl
+  | hapex b => rfl
+
+end Contraction
+
+def squareDen (s : unitInterval) : ℝ := max (s:ℝ) (1-(s:ℝ))
+lemma squareDen_pos (s : unitInterval) : 0 < squareDen s := by
+  have ha := le_max_left (s:ℝ) (1-(s:ℝ))
+  have hb := le_max_right (s:ℝ) (1-(s:ℝ))
+  dsimp [squareDen]
+  linarith
+lemma continuous_squareDen : Continuous squareDen :=
+  continuous_subtype_val.max (continuous_const.sub continuous_subtype_val)
+
+def horizontal : C(unitInterval × unitInterval,unitInterval) where
+  toFun st := ⟨(1-(st.1:ℝ))*(st.2:ℝ) / squareDen st.1,by
+    refine ⟨div_nonneg (mul_nonneg (sub_nonneg.mpr st.1.property.2) st.2.property.1)
+      (squareDen_pos st.1).le,?_⟩
+    apply (div_le_one (squareDen_pos st.1)).mpr
+    have h := mul_le_mul_of_nonneg_left st.2.property.2 (sub_nonneg.mpr st.1.property.2)
+    have h' : (1 - (st.1:ℝ)) * (st.2:ℝ) ≤ 1 - (st.1:ℝ) := by
+      simpa only [mul_one] using h
+    exact h'.trans (le_max_right _ _)⟩
+  continuous_toFun := by
+    apply Continuous.subtype_mk
+    exact (((continuous_const.sub (continuous_subtype_val.comp continuous_fst)).mul
+    (continuous_subtype_val.comp continuous_snd)).div
+      (continuous_squareDen.comp continuous_fst) (fun _ => (squareDen_pos _).ne'))
+
+def vertical : C(unitInterval × unitInterval,unitInterval) where
+  toFun st := ⟨(st.1:ℝ)*(st.2:ℝ) / squareDen st.1,by
+    refine ⟨div_nonneg (mul_nonneg st.1.property.1 st.2.property.1) (squareDen_pos st.1).le,?_⟩
+    apply (div_le_one (squareDen_pos st.1)).mpr
+    have h := mul_le_mul_of_nonneg_left st.2.property.2 st.1.property.1
+    have h' : (st.1:ℝ) * (st.2:ℝ) ≤ (st.1:ℝ) := by
+      simpa only [mul_one] using h
+    exact h'.trans (le_max_left _ _)⟩
+  continuous_toFun := by
+    apply Continuous.subtype_mk
+    exact (((continuous_subtype_val.comp continuous_fst).mul
+    (continuous_subtype_val.comp continuous_snd)).div
+      (continuous_squareDen.comp continuous_fst) (fun _ => (squareDen_pos _).ne'))
+
+@[simp] lemma horizontal_left (t : unitInterval) : horizontal (0,t) = t := by
+  apply Subtype.ext
+  simp [horizontal,squareDen]
+@[simp] lemma horizontal_right (t : unitInterval) : horizontal (1,t) = 0 := by
+  apply Subtype.ext
+  simp [horizontal,squareDen]
+@[simp] lemma vertical_left (t : unitInterval) : vertical (0,t) = 0 := by
+  apply Subtype.ext
+  simp [vertical,squareDen]
+@[simp] lemma vertical_right (t : unitInterval) : vertical (1,t) = t := by
+  apply Subtype.ext
+  simp [vertical,squareDen]
+@[simp] lemma horizontal_bottom (s : unitInterval) : horizontal (s,0) = 0 := by
+  apply Subtype.ext
+  simp [horizontal]
+@[simp] lemma vertical_bottom (s : unitInterval) : vertical (s,0) = 0 := by
+  apply Subtype.ext
+  simp [vertical]
+
+lemma square_top (s : unitInterval) : horizontal (s,1) = 1 ∨ vertical (s,1) = 1 := by
+  rcases le_total (s:ℝ) (1-(s:ℝ)) with hs | hs
+  · left
+    apply Subtype.ext
+    have hn : 1-(s:ℝ) ≠ 0 := by
+      have hd := squareDen_pos s
+      rw [squareDen,max_eq_right hs] at hd
+      exact hd.ne'
+    simp [horizontal,squareDen,max_eq_right hs,hn]
+  · right
+    apply Subtype.ext
+    have hn : (s:ℝ) ≠ 0 := by
+      have hd := squareDen_pos s
+      rw [squareDen,max_eq_left hs] at hd
+      exact hd.ne'
+    simp [vertical,squareDen,max_eq_left hs,hn]
+
+end NearbyLagrangian.FiberCone
+
+end
+
+noncomputable section
+universe u v w
+open Set ContinuousMap
+open scoped Topology unitInterval
+
+namespace NearbyLagrangian.FiberCone
+variable {B A X Y : Type*} [TopologicalSpace B] [TopologicalSpace A]
+  [TopologicalSpace X] [TopologicalSpace Y]
+
+def cylinderLift (f : C(A,X)) (q : C(A,B)) (i : C(unitInterval × X,Y))
+    (c : C(unitInterval × (unitInterval × A),Y)) (v : C(unitInterval × B,Y))
+    (h₀ : ∀ s a, c (s,(0,a)) = i (s,f a))
+    (h₁ : ∀ s a, c (s,(1,a)) = v (s,q a)) : C(unitInterval × Space f q,Y) :=
+  (ContinuousMap.uncurry (lift f q
+    (ContinuousMap.curry (i.comp ContinuousMap.prodSwap))
+    (ContinuousMap.curry (c.comp ContinuousMap.prodSwap))
+    (ContinuousMap.curry (v.comp ContinuousMap.prodSwap))
+    (fun a => by ext s; exact h₀ s a)
+    (fun a => by ext s; exact h₁ s a))).comp ContinuousMap.prodSwap
+
+@[simp] lemma cylinderLift_base (f : C(A,X)) (q : C(A,B)) (i : C(unitInterval × X,Y))
+    (c : C(unitInterval × (unitInterval × A),Y)) (v : C(unitInterval × B,Y))
+    (h₀ : ∀ s a, c (s,(0,a)) = i (s,f a))
+    (h₁ : ∀ s a, c (s,(1,a)) = v (s,q a)) (s : unitInterval) (x : X) :
+    cylinderLift f q i c v h₀ h₁ (s,base f q x) = i (s,x) := rfl
+@[simp] lemma cylinderLift_ray (f : C(A,X)) (q : C(A,B)) (i : C(unitInterval × X,Y))
+    (c : C(unitInterval × (unitInterval × A),Y)) (v : C(unitInterval × B,Y))
+    (h₀ : ∀ s a, c (s,(0,a)) = i (s,f a))
+    (h₁ : ∀ s a, c (s,(1,a)) = v (s,q a)) (s : unitInterval) (ta : unitInterval × A) :
+    cylinderLift f q i c v h₀ h₁ (s,ray f q ta) = c (s,ta) := rfl
+@[simp] lemma cylinderLift_apex (f : C(A,X)) (q : C(A,B)) (i : C(unitInterval × X,Y))
+    (c : C(unitInterval × (unitInterval × A),Y)) (v : C(unitInterval × B,Y))
+    (h₀ : ∀ s a, c (s,(0,a)) = i (s,f a))
+    (h₁ : ∀ s a, c (s,(1,a)) = v (s,q a)) (s : unitInterval) (b : B) :
+    cylinderLift f q i c v h₀ h₁ (s,apex f q b) = v (s,b) := rfl
+
+namespace Contraction
+variable {f : C(A,X)} {p : C(X,B)} {q : C(A,B)} {hf : ∀ a, p (f a) = q a}
+variable (H : Contraction f p q hf)
+
+def leftRay : C(unitInterval × (unitInterval × A),Susp q) where
+  toFun z := collapse f p q hf (H.hom
+    (vertical (z.1,z.2.1),ray f q (horizontal (z.1,z.2.1),z.2.2)))
+  continuous_toFun := (collapse f p q hf).continuous.comp (H.hom.continuous.comp
+    ((vertical.continuous.comp (continuous_fst.prodMk (continuous_fst.comp continuous_snd))).prodMk
+      ((ray f q).continuous.comp
+        ((horizontal.continuous.comp (continuous_fst.prodMk (continuous_fst.comp continuous_snd))).prodMk
+          (continuous_snd.comp continuous_snd)))))
+
+lemma leftRay_bottom (s : unitInterval) (a : A) : H.leftRay (s,(0,a)) = south q (q a) := by
+  change collapse f p q hf (H.hom (vertical (s,0),ray f q (horizontal (s,0),a))) = _
+  rw [horizontal_bottom,vertical_bottom,H.at_zero,ray_zero,collapse_base,hf]
+
+lemma leftRay_top (s : unitInterval) (a : A) : H.leftRay (s,(1,a)) = north q (q a) := by
+  change collapse f p q hf (H.hom (vertical (s,1),ray f q (horizontal (s,1),a))) = _
+  rcases square_top s with hx | hy
+  · rw [hx,ray_one,H.fixed,collapse_apex]
+  · rw [hy,H.at_one,projection_ray,collapse_apex]
+
+def leftDeformation : C(unitInterval × Susp q,Susp q) :=
+  cylinderLift q q ((south q).comp ⟨Prod.snd,continuous_snd⟩) H.leftRay
+    ((north q).comp ⟨Prod.snd,continuous_snd⟩) H.leftRay_bottom H.leftRay_top
+
+@[simp] lemma leftDeformation_south (s : unitInterval) (b : B) :
+    H.leftDeformation (s,south q b) = south q b := rfl
+@[simp] lemma leftDeformation_meridian (s : unitInterval) (ta : unitInterval × A) :
+    H.leftDeformation (s,meridian q ta) = H.leftRay (s,ta) := rfl
+@[simp] lemma leftDeformation_north (s : unitInterval) (b : B) :
+    H.leftDeformation (s,north q b) = north q b := rfl
+
+lemma leftDeformation_zero (z : Susp q) : H.leftDeformation (0,z) = z := by
+  induction z using induction with
+  | hbase b => rfl
+  | hray ta =>
+    rw [leftDeformation_meridian]
+    change collapse f p q hf (H.hom
+      (vertical (0,ta.1),ray f q (horizontal (0,ta.1),ta.2))) = _
+    rw [horizontal_left,vertical_left,H.at_zero,collapse_ray]
+  | hapex b => rfl
+
+lemma leftDeformation_one (z : Susp q) :
+    H.leftDeformation (1,z) = H.inverse (suspMap f p q hf z) := by
+  induction z using induction with
+  | hbase b => rw [leftDeformation_south,suspMap_south,inverse_south]
+  | hray ta =>
+    rw [leftDeformation_meridian,suspMap_meridian,inverse_meridian]
+    change collapse f p q hf (H.hom
+      (vertical (1,ta.1),ray f q (horizontal (1,ta.1),ta.2))) = _
+    rw [horizontal_right,vertical_right,ray_zero]
+  | hapex b => rw [leftDeformation_north,suspMap_north,inverse_north]
+
+lemma leftDeformation_over (s : unitInterval) (z : Susp q) :
+    suspProjection q (H.leftDeformation (s,z)) = suspProjection q z := by
+  induction z using induction with
+  | hbase b => rfl
+  | hray ta =>
+    rw [leftDeformation_meridian]
+    change suspProjection q (collapse f p q hf (H.hom
+      (vertical (s,ta.1),ray f q (horizontal (s,ta.1),ta.2)))) = _
+    rw [collapse_over,H.staysOver,projection_ray]
+    rfl
+  | hapex b => rfl
+
+def leftHomotopy : (H.inverse.comp (suspMap f p q hf)).Homotopy (ContinuousMap.id (Susp q)) :=
+  ContinuousMap.Homotopy.symm
+    { toContinuousMap := H.leftDeformation
+      map_zero_left := H.leftDeformation_zero
+      map_one_left := H.leftDeformation_one }
+
+lemma leftHomotopy_over (s : unitInterval) (z : Susp q) :
+    suspProjection q (H.leftHomotopy (s,z)) = suspProjection q z :=
+  H.leftDeformation_over (unitInterval.symm s) z
+
+end Contraction
+end NearbyLagrangian.FiberCone
+
+end
+
+end OAI

lean/OAI/Topology/Homotopy/FiberKernel.lean

@@ -1,57 +1,57 @@
-import Mathlib
-import OAI.Topology.Homotopy.HomotopyFiber
-import OAI.Topology.Homotopy.LoopHomotopy
-
-namespace OAI
-
-/-! Nontrivial homotopy kernels from finite cardinality bounds. -/
-
-noncomputable section
-universe u v w
-
-
-namespace NearbyLagrangian.HomotopyFiber
-open ContinuousMap
-open scoped Topology Topology.Homotopy
-open BasedHomotopy
-variable {N X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
-variable [DecidableEq N] [Nonempty N]
-
-theorem nontrivial_kernel_of_large_fiber (e : C(X,Y)) (x : X)
-    {K : Type*} [Fintype K]
-    [Fintype (HomotopyGroup N (Path (e x) (e x)) (Path.refl (e x)))]
-    (ι : K ↪ HomotopyGroup N (Fiber e (e x)) (base e x))
-    (hcard : Fintype.card (HomotopyGroup N (Path (e x) (e x)) (Path.refl (e x))) < Fintype.card K) :
-    ∃ a : HomotopyGroup N X x, a ≠ 1 ∧ groupMap e rfl a = 1 := by
-  classical
-  suffices h : ∃ k, groupMap (projection e (e x)) rfl (ι k) ≠ 1 by
-    obtain ⟨k,hk⟩ := h
-    exact ⟨groupMap (projection e (e x)) rfl (ι k),hk,
-      (kernel_eq_projection_range e x _).mpr ⟨ι k,rfl⟩⟩
-  by_contra! h
-  choose q hq using fun k => (projection_kernel_eq_loop_range e x (ι k)).mp (h k)
-  have hi : Function.Injective q := by
-    intro k l hkl
-    apply ι.injective
-    have hk := hq k
-    rw [hkl] at hk
-    exact hk.symm.trans (hq l)
-  exact (Nat.not_le_of_lt hcard) (Fintype.card_le_of_injective q hi)
-
-theorem nontrivial_pi9_kernel_of_source_sizes (e : C(X,Y)) (x : X)
-    (sphereCalculation : π_ 10 Y (e x) ≃ (Fin 3 → ZMod 2))
-    (whiteheadClasses : (ZMod 8 × (ZMod 2 × ZMod 2)) ↪
-      π_ 9 (Fiber e (e x)) (base e x)) :
-    ∃ a : π_ 9 X x, a ≠ 1 ∧ groupMap e rfl a = 1 := by
-  let calculation := (pi9LoopEquivPi10 (e x)).trans sphereCalculation
-  let : Fintype (π_ 9 (Path (e x) (e x)) (Path.refl (e x))) :=
-    Fintype.ofEquiv (Fin 3 → ZMod 2) calculation.symm
-  apply nontrivial_kernel_of_large_fiber e x whiteheadClasses
-  rw [Fintype.card_congr calculation]
-  norm_num
-
-end NearbyLagrangian.HomotopyFiber
-
-end
-
-end OAI
+import Mathlib
+import OAI.Topology.Homotopy.HomotopyFiber
+import OAI.Topology.Homotopy.LoopHomotopy
+
+namespace OAI
+
+/-! Nontrivial homotopy kernels from finite cardinality bounds. -/
+
+noncomputable section
+universe u v w
+
+
+namespace NearbyLagrangian.HomotopyFiber
+open ContinuousMap
+open scoped Topology Topology.Homotopy
+open BasedHomotopy
+variable {N X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+variable [DecidableEq N] [Nonempty N]
+
+theorem nontrivial_kernel_of_large_fiber (e : C(X,Y)) (x : X)
+    {K : Type*} [Fintype K]
+    [Fintype (HomotopyGroup N (Path (e x) (e x)) (Path.refl (e x)))]
+    (ι : K ↪ HomotopyGroup N (Fiber e (e x)) (base e x))
+    (hcard : Fintype.card (HomotopyGroup N (Path (e x) (e x)) (Path.refl (e x))) < Fintype.card K) :
+    ∃ a : HomotopyGroup N X x, a ≠ 1 ∧ groupMap e rfl a = 1 := by
+  classical
+  suffices h : ∃ k, groupMap (projection e (e x)) rfl (ι k) ≠ 1 by
+    obtain ⟨k,hk⟩ := h
+    exact ⟨groupMap (projection e (e x)) rfl (ι k),hk,
+      (kernel_eq_projection_range e x _).mpr ⟨ι k,rfl⟩⟩
+  by_contra! h
+  choose q hq using fun k => (projection_kernel_eq_loop_range e x (ι k)).mp (h k)
+  have hi : Function.Injective q := by
+    intro k l hkl
+    apply ι.injective
+    have hk := hq k
+    rw [hkl] at hk
+    exact hk.symm.trans (hq l)
+  exact (Nat.not_le_of_lt hcard) (Fintype.card_le_of_injective q hi)
+
+theorem nontrivial_pi9_kernel_of_source_sizes (e : C(X,Y)) (x : X)
+    (sphereCalculation : π_ 10 Y (e x) ≃ (Fin 3 → ZMod 2))
+    (whiteheadClasses : (ZMod 8 × (ZMod 2 × ZMod 2)) ↪
+      π_ 9 (Fiber e (e x)) (base e x)) :
+    ∃ a : π_ 9 X x, a ≠ 1 ∧ groupMap e rfl a = 1 := by
+  let calculation := (pi9LoopEquivPi10 (e x)).trans sphereCalculation
+  let : Fintype (π_ 9 (Path (e x) (e x)) (Path.refl (e x))) :=
+    Fintype.ofEquiv (Fin 3 → ZMod 2) calculation.symm
+  apply nontrivial_kernel_of_large_fiber e x whiteheadClasses
+  rw [Fintype.card_congr calculation]
+  norm_num
+
+end NearbyLagrangian.HomotopyFiber
+
+end
+
+end OAI

lean/OAI/Topology/Homotopy/HomotopyFiber.lean

@@ -1,319 +1,319 @@
-import Mathlib
-import OAI.Topology.Homotopy.BasedMaps
-
-namespace OAI
-
-/-! The path-space homotopy fiber and exactness at its projection. -/
-
-noncomputable section
-universe u v w
-
-
-namespace NearbyLagrangian.HomotopyFiber
-open ContinuousMap
-open scoped Topology Topology.Homotopy
-open BasedHomotopy
-variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
-
-def Fiber (e : C(X,Y)) (y : Y) :=
-  {z : X × C(unitInterval,Y) // z.2 0 = e z.1 ∧ z.2 1 = y}
-
-instance (e : C(X,Y)) (y : Y) : TopologicalSpace (Fiber e y) := inferInstanceAs
-  (TopologicalSpace {z : X × C(unitInterval,Y) // z.2 0 = e z.1 ∧ z.2 1 = y})
-
-def base (e : C(X,Y)) (x : X) : Fiber e (e x) :=
-  ⟨(x,ContinuousMap.const _ (e x)),rfl,rfl⟩
-
-def projection (e : C(X,Y)) (y : Y) : C(Fiber e y,X) :=
-  ⟨fun z => z.1.1,continuous_fst.comp continuous_subtype_val⟩
-
-@[simp] theorem projection_base (e : C(X,Y)) (x : X) :
-    projection e (e x) (base e x) = x := rfl
-
-variable {N : Type*}
-
-def projectionNullhomotopy (e : C(X,Y)) (x : X)
-    (f : GenLoop N (Fiber e (e x)) (base e x)) :
-    (loopMap e rfl (loopMap (projection e (e x)) rfl f)).1.HomotopyRel (ContinuousMap.const _ (e x)) (Cube.boundary N) where
-  toFun z := (f z.2).1.2 z.1
-  continuous_toFun := by
-    exact ((continuous_snd.comp (continuous_subtype_val.comp (f.1.continuous.comp continuous_snd))).eval continuous_fst)
-  map_zero_left t := (f t).2.1
-  map_one_left t := (f t).2.2
-  prop' s t ht := by
-    change (f t).1.2 s = e (f t).1.1
-    rw [GenLoop.boundary f t ht]
-    rfl
-
-def liftNullhomotopy (e : C(X,Y)) (x : X) (f : GenLoop N X x)
-    (H : (loopMap e rfl f).1.HomotopyRel (ContinuousMap.const _ (e x)) (Cube.boundary N)) :
-    GenLoop N (Fiber e (e x)) (base e x) :=
-  ⟨⟨fun t => ⟨(f t,(H.toContinuousMap.comp ContinuousMap.prodSwap).curry t),
-      H.map_zero_left t,H.map_one_left t⟩,by
-      apply Continuous.subtype_mk
-      exact f.1.continuous.prodMk (H.toContinuousMap.comp ContinuousMap.prodSwap).curry.continuous⟩,
-    by
-      intro t ht
-      apply Subtype.ext
-      apply Prod.ext (GenLoop.boundary f t ht)
-      ext s
-      exact (H.prop s t ht).trans (congrArg e (GenLoop.boundary f t ht))⟩
-
-@[simp] theorem projection_liftNullhomotopy (e : C(X,Y)) (x : X) (f : GenLoop N X x)
-    (H : (loopMap e rfl f).1.HomotopyRel (ContinuousMap.const _ (e x)) (Cube.boundary N)) :
-    loopMap (projection e (e x)) rfl (liftNullhomotopy e x f H) = f := by ext; rfl
-
-theorem kernel_eq_projection_range [DecidableEq N] [Nonempty N]
-    (e : C(X,Y)) (x : X) (a : HomotopyGroup N X x) :
-    groupMap e rfl a = 1 ↔
-      ∃ b : HomotopyGroup N (Fiber e (e x)) (base e x),
-        groupMap (projection e (e x)) rfl b = a := by
-  induction a using Quotient.inductionOn with
-  | h f =>
-    constructor
-    · intro h
-      change (⟦loopMap e rfl f⟧ : HomotopyGroup N Y (e x)) = ⟦GenLoop.const⟧ at h
-      obtain ⟨H⟩ := Quotient.exact h
-      exact ⟨⟦liftNullhomotopy e x f H⟧,congrArg Quotient.mk'' (projection_liftNullhomotopy e x f H)⟩
-    · rintro ⟨b,hb⟩
-      rw [← hb]
-      induction b using Quotient.inductionOn with
-      | h g =>
-        change (⟦loopMap e rfl (loopMap (projection e (e x)) rfl g)⟧ : HomotopyGroup N Y (e x)) = ⟦GenLoop.const⟧
-        exact Quotient.sound ⟨projectionNullhomotopy e x g⟩
-
-end NearbyLagrangian.HomotopyFiber
-
-end
-
-noncomputable section
-universe u v w
-
-
-namespace NearbyLagrangian.HomotopyFiber
-open ContinuousMap Set
-open scoped Topology Topology.Homotopy
-open BasedHomotopy
-
-def frontTime (s r : unitInterval) : unitInterval := projIcc 0 1 zero_le_one ((s:ℝ)-2*r)
-def backTime (s r : unitInterval) : unitInterval := projIcc 0 1 zero_le_one ((2*(r:ℝ)-s)/(2-s))
-
-lemma continuous_frontTime : Continuous (fun z : unitInterval × unitInterval => frontTime z.1 z.2) := by
-  unfold frontTime
-  fun_prop
-
-lemma continuous_backTime : Continuous (fun z : unitInterval × unitInterval => backTime z.1 z.2) := by
-  unfold backTime
-  apply continuous_projIcc.comp
-  apply Continuous.div
-  · fun_prop
-  · fun_prop
-  · intro z
-    have := z.1.property.2
-    linarith
-
-@[simp] lemma frontTime_zero_right (s : unitInterval) : frontTime s 0 = s := by
-  change projIcc 0 1 zero_le_one ((s:ℝ)-2*0) = s
-  simp only [mul_zero,sub_zero]
-  exact projIcc_val _ s
-
-@[simp] lemma backTime_one_right (s : unitInterval) : backTime s 1 = 1 := by
-  unfold backTime
-  have hs : (2:ℝ)-s ≠ 0 := by linarith [s.property.2]
-  simp [hs]
-
-@[simp] lemma backTime_zero_left (r : unitInterval) : backTime 0 r = r := by
-  change projIcc 0 1 zero_le_one ((2*(r:ℝ)-0)/(2-0)) = r
-  have he : (2*(r:ℝ)-(0:ℝ))/(2-(0:ℝ)) = r := by ring
-  rw [he]
-  exact projIcc_val _ r
-
-lemma frontTime_seam {s r : unitInterval} (h : 2*(r:ℝ)=s) : frontTime s r = 0 := by
-  unfold frontTime
-  rw [h,sub_self]
-  exact projIcc_left _
-
-lemma backTime_seam {s r : unitInterval} (h : 2*(r:ℝ)=s) : backTime s r = 0 := by
-  unfold backTime
-  rw [h,sub_self,zero_div]
-  exact projIcc_left _
-
-variable {A X Y : Type*} [TopologicalSpace A] [TopologicalSpace X] [TopologicalSpace Y]
-variable (e : C(X,Y)) (x : X) (f : C(A,Fiber e (e x)))
-variable (H : ((projection e (e x)).comp f).Homotopy (ContinuousMap.const A x))
-
-def deformValue (z : (unitInterval × A) × unitInterval) : Y :=
-  if 2*(z.2:ℝ) ≤ z.1.1 then e (H (frontTime z.1.1 z.2,z.1.2))
-  else (f z.1.2).1.2 (backTime z.1.1 z.2)
-
-lemma continuous_deformValue : Continuous (deformValue e x f H) := by
-  unfold deformValue
-  apply continuous_if_le
-  · fun_prop
-  · fun_prop
-  · apply Continuous.continuousOn
-    apply e.continuous.comp
-    apply H.continuous.comp
-    exact (continuous_frontTime.comp ((continuous_fst.comp continuous_fst).prodMk continuous_snd)).prodMk
-      (continuous_snd.comp continuous_fst)
-  · apply Continuous.continuousOn
-    exact (continuous_snd.comp (continuous_subtype_val.comp
-      (f.continuous.comp (continuous_snd.comp continuous_fst)))).eval
-        (continuous_backTime.comp ((continuous_fst.comp continuous_fst).prodMk continuous_snd))
-  · intro z hz
-    rw [frontTime_seam hz,backTime_seam hz,H.apply_zero]
-    exact (f z.1.2).2.1.symm
-
-@[simp] lemma deformValue_start (s : unitInterval) (a : A) :
-    deformValue e x f H ((s,a),0) = e (H (s,a)) := by
-  change (if 2*(0:ℝ) ≤ s then e (H (frontTime s 0,a)) else (f a).1.2 (backTime s 0)) = e (H (s,a))
-  simp only [mul_zero,ite_eq_left s.property.1,frontTime_zero_right]
-
-@[simp] lemma deformValue_end (s : unitInterval) (a : A) :
-    deformValue e x f H ((s,a),1) = e x := by
-  change (if 2*(1:ℝ) ≤ s then e (H (frontTime s 1,a)) else (f a).1.2 (backTime s 1)) = e x
-  have hs : ¬ 2*(1:ℝ) ≤ (s:ℝ) := by linarith [s.property.2]
-  rw [ite_eq_right hs,backTime_one_right]
-  exact (f a).2.2
-
-@[simp] lemma deformValue_zero (a : A) (r : unitInterval) :
-    deformValue e x f H ((0,a),r) = (f a).1.2 r := by
-  unfold deformValue
-  split_ifs with hr
-  · have he : r = 0 := Subtype.ext (by dsimp at hr ⊢; linarith [r.property.1])
-    rw [he,frontTime_zero_right,H.apply_zero]
-    exact (f a).2.1.symm
-  · rw [backTime_zero_left]
-
-def deformPath : C(unitInterval × A,C(unitInterval,Y)) :=
-  (⟨deformValue e x f H,continuous_deformValue e x f H⟩ : C((unitInterval × A) × unitInterval,Y)).curry
-
-def deformFiber : C(unitInterval × A,Fiber e (e x)) where
-  toFun z := ⟨(H z,deformPath e x f H z),deformValue_start e x f H z.1 z.2,deformValue_end e x f H z.1 z.2⟩
-  continuous_toFun := (H.continuous.prodMk (deformPath e x f H).continuous).subtype_mk _
-
-@[simp] lemma deformFiber_zero (a : A) : deformFiber e x f H (0,a) = f a := by
-  apply Subtype.ext
-  apply Prod.ext
-  · exact H.apply_zero a
-  · ext r
-    exact deformValue_zero e x f H a r
-
-def loopInclusion : C(Path (e x) (e x),Fiber e (e x)) where
-  toFun p := ⟨(x,p.toContinuousMap),p.source,p.target⟩
-  continuous_toFun := (continuous_const.prodMk continuous_induced_dom).subtype_mk _
-
-@[simp] lemma loopInclusion_base : loopInclusion e x (Path.refl (e x)) = base e x := rfl
-
-def loopRepresentative : C(A,Path (e x) (e x)) where
-  toFun a := {
-    toContinuousMap := deformPath e x f H (1,a)
-    source' := by
-      change deformValue e x f H ((1,a),0) = e x
-      rw [deformValue_start,H.apply_one,ContinuousMap.const_apply]
-    target' := deformValue_end e x f H 1 a }
-  continuous_toFun := by
-    apply Path.continuous_uncurry_iff.mp
-    exact (continuous_deformValue e x f H).comp ((continuous_const.prodMk continuous_fst).prodMk continuous_snd)
-
-def deformHomotopy : f.Homotopy ((loopInclusion e x).comp (loopRepresentative e x f H)) where
-  toContinuousMap := deformFiber e x f H
-  map_zero_left := deformFiber_zero e x f H
-  map_one_left a := by
-    apply Subtype.ext
-    apply Prod.ext
-    · exact H.apply_one a
-    · rfl
-
-lemma deformValue_fixed (a : A) (ha : f a = base e x) (hH : ∀ s, H (s,a) = x)
-    (s r : unitInterval) : deformValue e x f H ((s,a),r) = e x := by
-  unfold deformValue
-  split_ifs
-  · rw [hH]
-  · rw [ha]
-    rfl
-
-lemma deformFiber_fixed (a : A) (ha : f a = base e x) (hH : ∀ s, H (s,a) = x)
-    (s : unitInterval) : deformFiber e x f H (s,a) = base e x := by
-  apply Subtype.ext
-  apply Prod.ext (hH s)
-  ext r
-  exact deformValue_fixed e x f H a ha hH s r
-
-lemma loopRepresentative_fixed (a : A) (ha : f a = base e x) (hH : ∀ s, H (s,a) = x) :
-    loopRepresentative e x f H a = Path.refl (e x) := by
-  apply Path.ext
-  funext r
-  exact deformValue_fixed e x f H a ha hH 1 r
-
-end NearbyLagrangian.HomotopyFiber
-
-end
-
-noncomputable section
-universe u v w
-
-
-namespace NearbyLagrangian.HomotopyFiber
-open ContinuousMap
-open scoped Topology Topology.Homotopy
-open BasedHomotopy
-variable {N X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
-variable (e : C(X,Y)) (x : X)
-
-def boundaryRepresentative (f : GenLoop N (Fiber e (e x)) (base e x))
-    (H : (loopMap (projection e (e x)) rfl f).1.HomotopyRel
-      (ContinuousMap.const _ x) (Cube.boundary N)) :
-    GenLoop N (Path (e x) (e x)) (Path.refl (e x)) :=
-  ⟨loopRepresentative e x f.1 H.toHomotopy,by
-    intro t ht
-    apply loopRepresentative_fixed
-    · exact GenLoop.boundary f t ht
-    · intro s
-      exact (H.prop s t ht).trans (congrArg (projection e (e x)) (GenLoop.boundary f t ht))⟩
-
-def boundaryRepresentativeHomotopy (f : GenLoop N (Fiber e (e x)) (base e x))
-    (H : (loopMap (projection e (e x)) rfl f).1.HomotopyRel
-      (ContinuousMap.const _ x) (Cube.boundary N)) :
-    f.1.HomotopyRel (loopMap (loopInclusion e x) rfl (boundaryRepresentative e x f H)).1
-      (Cube.boundary N) where
-  toHomotopy := deformHomotopy e x f.1 H.toHomotopy
-  prop' s t ht := by
-    change deformFiber e x f.1 H.toHomotopy (s,t) = f t
-    rw [GenLoop.boundary f t ht]
-    apply deformFiber_fixed
-    · exact GenLoop.boundary f t ht
-    · intro r
-      exact (H.prop r t ht).trans (congrArg (projection e (e x)) (GenLoop.boundary f t ht))
-
-@[simp] theorem projection_loopInclusion (g : GenLoop N (Path (e x) (e x)) (Path.refl (e x))) :
-    loopMap (projection e (e x)) rfl (loopMap (loopInclusion e x) rfl g) = GenLoop.const := by
-  ext t
-  rfl
-
-theorem projection_kernel_eq_loop_range [DecidableEq N] [Nonempty N]
-    (a : HomotopyGroup N (Fiber e (e x)) (base e x)) :
-    groupMap (projection e (e x)) rfl a = 1 ↔
-      ∃ b : HomotopyGroup N (Path (e x) (e x)) (Path.refl (e x)),
-        groupMap (loopInclusion e x) rfl b = a := by
-  induction a using Quotient.inductionOn with
-  | h f =>
-    constructor
-    · intro h
-      change (⟦loopMap (projection e (e x)) rfl f⟧ : HomotopyGroup N X x) = ⟦GenLoop.const⟧ at h
-      obtain ⟨H⟩ := Quotient.exact h
-      refine ⟨⟦boundaryRepresentative e x f H⟧, ?_⟩
-      exact Quotient.sound ⟨(boundaryRepresentativeHomotopy e x f H).symm⟩
-    · rintro ⟨b,hb⟩
-      rw [← hb]
-      induction b using Quotient.inductionOn with
-      | h g =>
-        change (⟦loopMap (projection e (e x)) rfl (loopMap (loopInclusion e x) rfl g)⟧ :
-          HomotopyGroup N X x) = ⟦GenLoop.const⟧
-        exact congrArg Quotient.mk'' (projection_loopInclusion e x g)
-
-end NearbyLagrangian.HomotopyFiber
-
-end
-
-end OAI
+import Mathlib
+import OAI.Topology.Homotopy.BasedMaps
+
+namespace OAI
+
+/-! The path-space homotopy fiber and exactness at its projection. -/
+
+noncomputable section
+universe u v w
+
+
+namespace NearbyLagrangian.HomotopyFiber
+open ContinuousMap
+open scoped Topology Topology.Homotopy
+open BasedHomotopy
+variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+
+def Fiber (e : C(X,Y)) (y : Y) :=
+  {z : X × C(unitInterval,Y) // z.2 0 = e z.1 ∧ z.2 1 = y}
+
+instance (e : C(X,Y)) (y : Y) : TopologicalSpace (Fiber e y) := inferInstanceAs
+  (TopologicalSpace {z : X × C(unitInterval,Y) // z.2 0 = e z.1 ∧ z.2 1 = y})
+
+def base (e : C(X,Y)) (x : X) : Fiber e (e x) :=
+  ⟨(x,ContinuousMap.const _ (e x)),rfl,rfl⟩
+
+def projection (e : C(X,Y)) (y : Y) : C(Fiber e y,X) :=
+  ⟨fun z => z.1.1,continuous_fst.comp continuous_subtype_val⟩
+
+@[simp] theorem projection_base (e : C(X,Y)) (x : X) :
+    projection e (e x) (base e x) = x := rfl
+
+variable {N : Type*}
+
+def projectionNullhomotopy (e : C(X,Y)) (x : X)
+    (f : GenLoop N (Fiber e (e x)) (base e x)) :
+    (loopMap e rfl (loopMap (projection e (e x)) rfl f)).1.HomotopyRel (ContinuousMap.const _ (e x)) (Cube.boundary N) where
+  toFun z := (f z.2).1.2 z.1
+  continuous_toFun := by
+    exact ((continuous_snd.comp (continuous_subtype_val.comp (f.1.continuous.comp continuous_snd))).eval continuous_fst)
+  map_zero_left t := (f t).2.1
+  map_one_left t := (f t).2.2
+  prop' s t ht := by
+    change (f t).1.2 s = e (f t).1.1
+    rw [GenLoop.boundary f t ht]
+    rfl
+
+def liftNullhomotopy (e : C(X,Y)) (x : X) (f : GenLoop N X x)
+    (H : (loopMap e rfl f).1.HomotopyRel (ContinuousMap.const _ (e x)) (Cube.boundary N)) :
+    GenLoop N (Fiber e (e x)) (base e x) :=
+  ⟨⟨fun t => ⟨(f t,(H.toContinuousMap.comp ContinuousMap.prodSwap).curry t),
+      H.map_zero_left t,H.map_one_left t⟩,by
+      apply Continuous.subtype_mk
+      exact f.1.continuous.prodMk (H.toContinuousMap.comp ContinuousMap.prodSwap).curry.continuous⟩,
+    by
+      intro t ht
+      apply Subtype.ext
+      apply Prod.ext (GenLoop.boundary f t ht)
+      ext s
+      exact (H.prop s t ht).trans (congrArg e (GenLoop.boundary f t ht))⟩
+
+@[simp] theorem projection_liftNullhomotopy (e : C(X,Y)) (x : X) (f : GenLoop N X x)
+    (H : (loopMap e rfl f).1.HomotopyRel (ContinuousMap.const _ (e x)) (Cube.boundary N)) :
+    loopMap (projection e (e x)) rfl (liftNullhomotopy e x f H) = f := by ext; rfl
+
+theorem kernel_eq_projection_range [DecidableEq N] [Nonempty N]
+    (e : C(X,Y)) (x : X) (a : HomotopyGroup N X x) :
+    groupMap e rfl a = 1 ↔
+      ∃ b : HomotopyGroup N (Fiber e (e x)) (base e x),
+        groupMap (projection e (e x)) rfl b = a := by
+  induction a using Quotient.inductionOn with
+  | h f =>
+    constructor
+    · intro h
+      change (⟦loopMap e rfl f⟧ : HomotopyGroup N Y (e x)) = ⟦GenLoop.const⟧ at h
+      obtain ⟨H⟩ := Quotient.exact h
+      exact ⟨⟦liftNullhomotopy e x f H⟧,congrArg Quotient.mk'' (projection_liftNullhomotopy e x f H)⟩
+    · rintro ⟨b,hb⟩
+      rw [← hb]
+      induction b using Quotient.inductionOn with
+      | h g =>
+        change (⟦loopMap e rfl (loopMap (projection e (e x)) rfl g)⟧ : HomotopyGroup N Y (e x)) = ⟦GenLoop.const⟧
+        exact Quotient.sound ⟨projectionNullhomotopy e x g⟩
+
+end NearbyLagrangian.HomotopyFiber
+
+end
+
+noncomputable section
+universe u v w
+
+
+namespace NearbyLagrangian.HomotopyFiber
+open ContinuousMap Set
+open scoped Topology Topology.Homotopy
+open BasedHomotopy
+
+def frontTime (s r : unitInterval) : unitInterval := projIcc 0 1 zero_le_one ((s:ℝ)-2*r)
+def backTime (s r : unitInterval) : unitInterval := projIcc 0 1 zero_le_one ((2*(r:ℝ)-s)/(2-s))
+
+lemma continuous_frontTime : Continuous (fun z : unitInterval × unitInterval => frontTime z.1 z.2) := by
+  unfold frontTime
+  fun_prop
+
+lemma continuous_backTime : Continuous (fun z : unitInterval × unitInterval => backTime z.1 z.2) := by
+  unfold backTime
+  apply continuous_projIcc.comp
+  apply Continuous.div
+  · fun_prop
+  · fun_prop
+  · intro z
+    have := z.1.property.2
+    linarith
+
+@[simp] lemma frontTime_zero_right (s : unitInterval) : frontTime s 0 = s := by
+  change projIcc 0 1 zero_le_one ((s:ℝ)-2*0) = s
+  simp only [mul_zero,sub_zero]
+  exact projIcc_val _ s
+
+@[simp] lemma backTime_one_right (s : unitInterval) : backTime s 1 = 1 := by
+  unfold backTime
+  have hs : (2:ℝ)-s ≠ 0 := by linarith [s.property.2]
+  simp [hs]
+
+@[simp] lemma backTime_zero_left (r : unitInterval) : backTime 0 r = r := by
+  change projIcc 0 1 zero_le_one ((2*(r:ℝ)-0)/(2-0)) = r
+  have he : (2*(r:ℝ)-(0:ℝ))/(2-(0:ℝ)) = r := by ring
+  rw [he]
+  exact projIcc_val _ r
+
+lemma frontTime_seam {s r : unitInterval} (h : 2*(r:ℝ)=s) : frontTime s r = 0 := by
+  unfold frontTime
+  rw [h,sub_self]
+  exact projIcc_left _
+
+lemma backTime_seam {s r : unitInterval} (h : 2*(r:ℝ)=s) : backTime s r = 0 := by
+  unfold backTime
+  rw [h,sub_self,zero_div]
+  exact projIcc_left _
+
+variable {A X Y : Type*} [TopologicalSpace A] [TopologicalSpace X] [TopologicalSpace Y]
+variable (e : C(X,Y)) (x : X) (f : C(A,Fiber e (e x)))
+variable (H : ((projection e (e x)).comp f).Homotopy (ContinuousMap.const A x))
+
+def deformValue (z : (unitInterval × A) × unitInterval) : Y :=
+  if 2*(z.2:ℝ) ≤ z.1.1 then e (H (frontTime z.1.1 z.2,z.1.2))
+  else (f z.1.2).1.2 (backTime z.1.1 z.2)
+
+lemma continuous_deformValue : Continuous (deformValue e x f H) := by
+  unfold deformValue
+  apply continuous_if_le
+  · fun_prop
+  · fun_prop
+  · apply Continuous.continuousOn
+    apply e.continuous.comp
+    apply H.continuous.comp
+    exact (continuous_frontTime.comp ((continuous_fst.comp continuous_fst).prodMk continuous_snd)).prodMk
+      (continuous_snd.comp continuous_fst)
+  · apply Continuous.continuousOn
+    exact (continuous_snd.comp (continuous_subtype_val.comp
+      (f.continuous.comp (continuous_snd.comp continuous_fst)))).eval
+        (continuous_backTime.comp ((continuous_fst.comp continuous_fst).prodMk continuous_snd))
+  · intro z hz
+    rw [frontTime_seam hz,backTime_seam hz,H.apply_zero]
+    exact (f z.1.2).2.1.symm
+
+@[simp] lemma deformValue_start (s : unitInterval) (a : A) :
+    deformValue e x f H ((s,a),0) = e (H (s,a)) := by
+  change (if 2*(0:ℝ) ≤ s then e (H (frontTime s 0,a)) else (f a).1.2 (backTime s 0)) = e (H (s,a))
+  simp only [mul_zero,ite_eq_left s.property.1,frontTime_zero_right]
+
+@[simp] lemma deformValue_end (s : unitInterval) (a : A) :
+    deformValue e x f H ((s,a),1) = e x := by
+  change (if 2*(1:ℝ) ≤ s then e (H (frontTime s 1,a)) else (f a).1.2 (backTime s 1)) = e x
+  have hs : ¬ 2*(1:ℝ) ≤ (s:ℝ) := by linarith [s.property.2]
+  rw [ite_eq_right hs,backTime_one_right]
+  exact (f a).2.2
+
+@[simp] lemma deformValue_zero (a : A) (r : unitInterval) :
+    deformValue e x f H ((0,a),r) = (f a).1.2 r := by
+  unfold deformValue
+  split_ifs with hr
+  · have he : r = 0 := Subtype.ext (by dsimp at hr ⊢; linarith [r.property.1])
+    rw [he,frontTime_zero_right,H.apply_zero]
+    exact (f a).2.1.symm
+  · rw [backTime_zero_left]
+
+def deformPath : C(unitInterval × A,C(unitInterval,Y)) :=
+  (⟨deformValue e x f H,continuous_deformValue e x f H⟩ : C((unitInterval × A) × unitInterval,Y)).curry
+
+def deformFiber : C(unitInterval × A,Fiber e (e x)) where
+  toFun z := ⟨(H z,deformPath e x f H z),deformValue_start e x f H z.1 z.2,deformValue_end e x f H z.1 z.2⟩
+  continuous_toFun := (H.continuous.prodMk (deformPath e x f H).continuous).subtype_mk _
+
+@[simp] lemma deformFiber_zero (a : A) : deformFiber e x f H (0,a) = f a := by
+  apply Subtype.ext
+  apply Prod.ext
+  · exact H.apply_zero a
+  · ext r
+    exact deformValue_zero e x f H a r
+
+def loopInclusion : C(Path (e x) (e x),Fiber e (e x)) where
+  toFun p := ⟨(x,p.toContinuousMap),p.source,p.target⟩
+  continuous_toFun := (continuous_const.prodMk continuous_induced_dom).subtype_mk _
+
+@[simp] lemma loopInclusion_base : loopInclusion e x (Path.refl (e x)) = base e x := rfl
+
+def loopRepresentative : C(A,Path (e x) (e x)) where
+  toFun a := {
+    toContinuousMap := deformPath e x f H (1,a)
+    source' := by
+      change deformValue e x f H ((1,a),0) = e x
+      rw [deformValue_start,H.apply_one,ContinuousMap.const_apply]
+    target' := deformValue_end e x f H 1 a }
+  continuous_toFun := by
+    apply Path.continuous_uncurry_iff.mp
+    exact (continuous_deformValue e x f H).comp ((continuous_const.prodMk continuous_fst).prodMk continuous_snd)
+
+def deformHomotopy : f.Homotopy ((loopInclusion e x).comp (loopRepresentative e x f H)) where
+  toContinuousMap := deformFiber e x f H
+  map_zero_left := deformFiber_zero e x f H
+  map_one_left a := by
+    apply Subtype.ext
+    apply Prod.ext
+    · exact H.apply_one a
+    · rfl
+
+lemma deformValue_fixed (a : A) (ha : f a = base e x) (hH : ∀ s, H (s,a) = x)
+    (s r : unitInterval) : deformValue e x f H ((s,a),r) = e x := by
+  unfold deformValue
+  split_ifs
+  · rw [hH]
+  · rw [ha]
+    rfl
+
+lemma deformFiber_fixed (a : A) (ha : f a = base e x) (hH : ∀ s, H (s,a) = x)
+    (s : unitInterval) : deformFiber e x f H (s,a) = base e x := by
+  apply Subtype.ext
+  apply Prod.ext (hH s)
+  ext r
+  exact deformValue_fixed e x f H a ha hH s r
+
+lemma loopRepresentative_fixed (a : A) (ha : f a = base e x) (hH : ∀ s, H (s,a) = x) :
+    loopRepresentative e x f H a = Path.refl (e x) := by
+  apply Path.ext
+  funext r
+  exact deformValue_fixed e x f H a ha hH 1 r
+
+end NearbyLagrangian.HomotopyFiber
+
+end
+
+noncomputable section
+universe u v w
+
+
+namespace NearbyLagrangian.HomotopyFiber
+open ContinuousMap
+open scoped Topology Topology.Homotopy
+open BasedHomotopy
+variable {N X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+variable (e : C(X,Y)) (x : X)
+
+def boundaryRepresentative (f : GenLoop N (Fiber e (e x)) (base e x))
+    (H : (loopMap (projection e (e x)) rfl f).1.HomotopyRel
+      (ContinuousMap.const _ x) (Cube.boundary N)) :
+    GenLoop N (Path (e x) (e x)) (Path.refl (e x)) :=
+  ⟨loopRepresentative e x f.1 H.toHomotopy,by
+    intro t ht
+    apply loopRepresentative_fixed
+    · exact GenLoop.boundary f t ht
+    · intro s
+      exact (H.prop s t ht).trans (congrArg (projection e (e x)) (GenLoop.boundary f t ht))⟩
+
+def boundaryRepresentativeHomotopy (f : GenLoop N (Fiber e (e x)) (base e x))
+    (H : (loopMap (projection e (e x)) rfl f).1.HomotopyRel
+      (ContinuousMap.const _ x) (Cube.boundary N)) :
+    f.1.HomotopyRel (loopMap (loopInclusion e x) rfl (boundaryRepresentative e x f H)).1
+      (Cube.boundary N) where
+  toHomotopy := deformHomotopy e x f.1 H.toHomotopy
+  prop' s t ht := by
+    change deformFiber e x f.1 H.toHomotopy (s,t) = f t
+    rw [GenLoop.boundary f t ht]
+    apply deformFiber_fixed
+    · exact GenLoop.boundary f t ht
+    · intro r
+      exact (H.prop r t ht).trans (congrArg (projection e (e x)) (GenLoop.boundary f t ht))
+
+@[simp] theorem projection_loopInclusion (g : GenLoop N (Path (e x) (e x)) (Path.refl (e x))) :
+    loopMap (projection e (e x)) rfl (loopMap (loopInclusion e x) rfl g) = GenLoop.const := by
+  ext t
+  rfl
+
+theorem projection_kernel_eq_loop_range [DecidableEq N] [Nonempty N]
+    (a : HomotopyGroup N (Fiber e (e x)) (base e x)) :
+    groupMap (projection e (e x)) rfl a = 1 ↔
+      ∃ b : HomotopyGroup N (Path (e x) (e x)) (Path.refl (e x)),
+        groupMap (loopInclusion e x) rfl b = a := by
+  induction a using Quotient.inductionOn with
+  | h f =>
+    constructor
+    · intro h
+      change (⟦loopMap (projection e (e x)) rfl f⟧ : HomotopyGroup N X x) = ⟦GenLoop.const⟧ at h
+      obtain ⟨H⟩ := Quotient.exact h
+      refine ⟨⟦boundaryRepresentative e x f H⟧, ?_⟩
+      exact Quotient.sound ⟨(boundaryRepresentativeHomotopy e x f H).symm⟩
+    · rintro ⟨b,hb⟩
+      rw [← hb]
+      induction b using Quotient.inductionOn with
+      | h g =>
+        change (⟦loopMap (projection e (e x)) rfl (loopMap (loopInclusion e x) rfl g)⟧ :
+          HomotopyGroup N X x) = ⟦GenLoop.const⟧
+        exact congrArg Quotient.mk'' (projection_loopInclusion e x g)
+
+end NearbyLagrangian.HomotopyFiber
+
+end
+
+end OAI

lean/OAI/Topology/Homotopy/LoopHomotopy.lean

@@ -1,143 +1,143 @@
-import Mathlib
-
-namespace OAI
-
-/-! Loop-space homeomorphisms and the degree-shift equivalence. -/
-
-noncomputable section
-universe u v w
-
-
-namespace NearbyLagrangian.BasedHomotopy
-open ContinuousMap
-open scoped Topology Topology.Homotopy
-variable {N M X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
-variable {x : X} {y : Y}
-
-theorem genLoop_homotopic_iff_joined (f g : GenLoop N X x) :
-    GenLoop.Homotopic f g ↔ Joined f g := by
-  constructor
-  · rintro ⟨H⟩
-    refine ⟨{
-      toFun := fun s => ⟨H.toContinuousMap.curry s,fun t ht =>
-        (H.prop s t ht).trans (GenLoop.boundary f t ht)⟩
-      continuous_toFun := H.toContinuousMap.curry.continuous.subtype_mk _
-      source' := ?_
-      target' := ?_ }⟩
-    · ext t; exact H.apply_zero t
-    · ext t; exact H.apply_one t
-  · rintro ⟨P⟩
-    refine ⟨{
-      toFun := fun z => P z.1 z.2
-      continuous_toFun := ?_
-      map_zero_left := ?_
-      map_one_left := ?_
-      prop' := ?_ }⟩
-    · exact (continuous_subtype_val.comp (P.continuous.comp continuous_fst)).eval continuous_snd
-    · intro t; exact congrArg (fun z : GenLoop N X x => z t) P.source
-    · intro t; exact congrArg (fun z : GenLoop N X x => z t) P.target
-    · intro s t ht
-      exact (GenLoop.boundary (P s) t ht).trans (GenLoop.boundary f t ht).symm
-
-def piEquivOfLoopHomeomorph (e : GenLoop N X x ≃ₜ GenLoop M Y y) :
-    HomotopyGroup N X x ≃ HomotopyGroup M Y y :=
-  Quotient.congr e.toEquiv fun f g => by
-    change GenLoop.Homotopic f g ↔ GenLoop.Homotopic (e f) (e g)
-    rw [genLoop_homotopic_iff_joined,genLoop_homotopic_iff_joined]
-    constructor
-    · intro h; exact h.map e.continuous_toFun
-    · intro h
-      have hh : Joined (e.symm (e f)) (e.symm (e g)) := h.map e.symm.continuous_toFun
-      simpa only [Homeomorph.symm_apply_apply] using hh
-
-def piReindexEquiv (e : N ≃ M) (x : X) :
-    HomotopyGroup N X x ≃ HomotopyGroup M X x :=
-  piEquivOfLoopHomeomorph (GenLoop.congr x e)
-
-def flattenPathLoop (f : GenLoop N (Path y y) (Path.refl y)) :
-    GenLoop (N ⊕ Unit) Y y :=
-  ⟨⟨fun a => f (a ∘ Sum.inl) (a (Sum.inr ())),by
-    apply Continuous.eval
-    · exact f.1.continuous.comp (by fun_prop)
-    · fun_prop⟩,by
-      rintro a ⟨i | i,hi⟩ <;> change f (a ∘ Sum.inl) (a (Sum.inr ())) = y
-      · have h : a ∘ Sum.inl ∈ Cube.boundary N := ⟨i,hi⟩
-        rw [GenLoop.boundary f _ h]
-        rfl
-      · cases i
-        rcases hi with hi | hi
-        · rw [hi]; exact (f _).source
-        · rw [hi]; exact (f _).target⟩
-
-def curryPathLoop (g : GenLoop (N ⊕ Unit) Y y) :
-    GenLoop N (Path y y) (Path.refl y) :=
-  ⟨⟨fun a => {
-    toFun := fun r => g (Sum.elim a (fun _ => r))
-    continuous_toFun := g.1.continuous.comp (by apply continuous_pi; intro i; cases i <;> dsimp only [Sum.elim] <;> fun_prop)
-    source' := GenLoop.boundary g _ ⟨Sum.inr (),Or.inl rfl⟩
-    target' := GenLoop.boundary g _ ⟨Sum.inr (),Or.inr rfl⟩ },by
-      apply Path.continuous_uncurry_iff.mp
-      exact g.1.continuous.comp (by apply continuous_pi; intro i; cases i <;> dsimp only [Sum.elim] <;> fun_prop)⟩,by
-    rintro a ⟨i,hi⟩
-    apply Path.ext
-    funext r
-    exact GenLoop.boundary g _ ⟨Sum.inl i,hi⟩⟩
-
-@[simp] lemma curry_flatten (f : GenLoop N (Path y y) (Path.refl y)) :
-    curryPathLoop (flattenPathLoop f) = f := by
-  apply Subtype.ext
-  ext a r
-  rfl
-
-@[simp] lemma flatten_curry (g : GenLoop (N ⊕ Unit) Y y) :
-    flattenPathLoop (curryPathLoop g) = g := by
-  apply Subtype.ext
-  ext a
-  change g (Sum.elim (a ∘ Sum.inl) (fun _ => a (Sum.inr ()))) = g a
-  congr 1
-  funext i
-  cases i with
-  | inl i => rfl
-  | inr i => cases i; rfl
-
-lemma continuous_flattenPathLoop : Continuous (@flattenPathLoop N Y _ y) := by
-  apply Continuous.subtype_mk
-  apply ContinuousMap.continuous_of_continuous_uncurry
-  change Continuous (fun z : GenLoop N (Path y y) (Path.refl y) × ((N ⊕ Unit) → unitInterval) =>
-    z.1 (z.2 ∘ Sum.inl) (z.2 (Sum.inr ())))
-  have h : Continuous (fun z : GenLoop N (Path y y) (Path.refl y) × ((N ⊕ Unit) → unitInterval) =>
-      z.1 (z.2 ∘ Sum.inl)) :=
-    (continuous_subtype_val.comp continuous_fst).eval
-      (continuous_pi (fun i => (continuous_apply (Sum.inl i)).comp continuous_snd))
-  exact h.eval ((continuous_apply (Sum.inr ())).comp continuous_snd)
-
-lemma continuous_curryPathLoop : Continuous (@curryPathLoop N Y _ y) := by
-  apply Continuous.subtype_mk
-  apply ContinuousMap.continuous_of_continuous_uncurry
-  apply Path.continuous_uncurry_iff.mp
-  exact (continuous_subtype_val.comp (continuous_fst.comp continuous_fst)).eval
-    (by apply continuous_pi; intro i; cases i <;> dsimp only [Sum.elim] <;> fun_prop)
-
-def loopSpaceCubeHomeomorph :
-    GenLoop N (Path y y) (Path.refl y) ≃ₜ GenLoop (N ⊕ Unit) Y y where
-  toFun := flattenPathLoop
-  invFun := curryPathLoop
-  left_inv := curry_flatten
-  right_inv := flatten_curry
-  continuous_toFun := continuous_flattenPathLoop
-  continuous_invFun := continuous_curryPathLoop
-
-def loopPiEquiv (y : Y) :
-    HomotopyGroup N (Path y y) (Path.refl y) ≃ HomotopyGroup (N ⊕ Unit) Y y :=
-  piEquivOfLoopHomeomorph loopSpaceCubeHomeomorph
-
-def pi9LoopEquivPi10 (y : Y) :
-    π_ 9 (Path y y) (Path.refl y) ≃ π_ 10 Y y :=
-  (loopPiEquiv y).trans (piReindexEquiv
-    ((Equiv.sumCongr (Equiv.refl (Fin 9)) (Equiv.ofUnique Unit (Fin 1))).trans finSumFinEquiv) y)
-
-end NearbyLagrangian.BasedHomotopy
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Loop-space homeomorphisms and the degree-shift equivalence. -/
+
+noncomputable section
+universe u v w
+
+
+namespace NearbyLagrangian.BasedHomotopy
+open ContinuousMap
+open scoped Topology Topology.Homotopy
+variable {N M X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+variable {x : X} {y : Y}
+
+theorem genLoop_homotopic_iff_joined (f g : GenLoop N X x) :
+    GenLoop.Homotopic f g ↔ Joined f g := by
+  constructor
+  · rintro ⟨H⟩
+    refine ⟨{
+      toFun := fun s => ⟨H.toContinuousMap.curry s,fun t ht =>
+        (H.prop s t ht).trans (GenLoop.boundary f t ht)⟩
+      continuous_toFun := H.toContinuousMap.curry.continuous.subtype_mk _
+      source' := ?_
+      target' := ?_ }⟩
+    · ext t; exact H.apply_zero t
+    · ext t; exact H.apply_one t
+  · rintro ⟨P⟩
+    refine ⟨{
+      toFun := fun z => P z.1 z.2
+      continuous_toFun := ?_
+      map_zero_left := ?_
+      map_one_left := ?_
+      prop' := ?_ }⟩
+    · exact (continuous_subtype_val.comp (P.continuous.comp continuous_fst)).eval continuous_snd
+    · intro t; exact congrArg (fun z : GenLoop N X x => z t) P.source
+    · intro t; exact congrArg (fun z : GenLoop N X x => z t) P.target
+    · intro s t ht
+      exact (GenLoop.boundary (P s) t ht).trans (GenLoop.boundary f t ht).symm
+
+def piEquivOfLoopHomeomorph (e : GenLoop N X x ≃ₜ GenLoop M Y y) :
+    HomotopyGroup N X x ≃ HomotopyGroup M Y y :=
+  Quotient.congr e.toEquiv fun f g => by
+    change GenLoop.Homotopic f g ↔ GenLoop.Homotopic (e f) (e g)
+    rw [genLoop_homotopic_iff_joined,genLoop_homotopic_iff_joined]
+    constructor
+    · intro h; exact h.map e.continuous_toFun
+    · intro h
+      have hh : Joined (e.symm (e f)) (e.symm (e g)) := h.map e.symm.continuous_toFun
+      simpa only [Homeomorph.symm_apply_apply] using hh
+
+def piReindexEquiv (e : N ≃ M) (x : X) :
+    HomotopyGroup N X x ≃ HomotopyGroup M X x :=
+  piEquivOfLoopHomeomorph (GenLoop.congr x e)
+
+def flattenPathLoop (f : GenLoop N (Path y y) (Path.refl y)) :
+    GenLoop (N ⊕ Unit) Y y :=
+  ⟨⟨fun a => f (a ∘ Sum.inl) (a (Sum.inr ())),by
+    apply Continuous.eval
+    · exact f.1.continuous.comp (by fun_prop)
+    · fun_prop⟩,by
+      rintro a ⟨i | i,hi⟩ <;> change f (a ∘ Sum.inl) (a (Sum.inr ())) = y
+      · have h : a ∘ Sum.inl ∈ Cube.boundary N := ⟨i,hi⟩
+        rw [GenLoop.boundary f _ h]
+        rfl
+      · cases i
+        rcases hi with hi | hi
+        · rw [hi]; exact (f _).source
+        · rw [hi]; exact (f _).target⟩
+
+def curryPathLoop (g : GenLoop (N ⊕ Unit) Y y) :
+    GenLoop N (Path y y) (Path.refl y) :=
+  ⟨⟨fun a => {
+    toFun := fun r => g (Sum.elim a (fun _ => r))
+    continuous_toFun := g.1.continuous.comp (by apply continuous_pi; intro i; cases i <;> dsimp only [Sum.elim] <;> fun_prop)
+    source' := GenLoop.boundary g _ ⟨Sum.inr (),Or.inl rfl⟩
+    target' := GenLoop.boundary g _ ⟨Sum.inr (),Or.inr rfl⟩ },by
+      apply Path.continuous_uncurry_iff.mp
+      exact g.1.continuous.comp (by apply continuous_pi; intro i; cases i <;> dsimp only [Sum.elim] <;> fun_prop)⟩,by
+    rintro a ⟨i,hi⟩
+    apply Path.ext
+    funext r
+    exact GenLoop.boundary g _ ⟨Sum.inl i,hi⟩⟩
+
+@[simp] lemma curry_flatten (f : GenLoop N (Path y y) (Path.refl y)) :
+    curryPathLoop (flattenPathLoop f) = f := by
+  apply Subtype.ext
+  ext a r
+  rfl
+
+@[simp] lemma flatten_curry (g : GenLoop (N ⊕ Unit) Y y) :
+    flattenPathLoop (curryPathLoop g) = g := by
+  apply Subtype.ext
+  ext a
+  change g (Sum.elim (a ∘ Sum.inl) (fun _ => a (Sum.inr ()))) = g a
+  congr 1
+  funext i
+  cases i with
+  | inl i => rfl
+  | inr i => cases i; rfl
+
+lemma continuous_flattenPathLoop : Continuous (@flattenPathLoop N Y _ y) := by
+  apply Continuous.subtype_mk
+  apply ContinuousMap.continuous_of_continuous_uncurry
+  change Continuous (fun z : GenLoop N (Path y y) (Path.refl y) × ((N ⊕ Unit) → unitInterval) =>
+    z.1 (z.2 ∘ Sum.inl) (z.2 (Sum.inr ())))
+  have h : Continuous (fun z : GenLoop N (Path y y) (Path.refl y) × ((N ⊕ Unit) → unitInterval) =>
+      z.1 (z.2 ∘ Sum.inl)) :=
+    (continuous_subtype_val.comp continuous_fst).eval
+      (continuous_pi (fun i => (continuous_apply (Sum.inl i)).comp continuous_snd))
+  exact h.eval ((continuous_apply (Sum.inr ())).comp continuous_snd)
+
+lemma continuous_curryPathLoop : Continuous (@curryPathLoop N Y _ y) := by
+  apply Continuous.subtype_mk
+  apply ContinuousMap.continuous_of_continuous_uncurry
+  apply Path.continuous_uncurry_iff.mp
+  exact (continuous_subtype_val.comp (continuous_fst.comp continuous_fst)).eval
+    (by apply continuous_pi; intro i; cases i <;> dsimp only [Sum.elim] <;> fun_prop)
+
+def loopSpaceCubeHomeomorph :
+    GenLoop N (Path y y) (Path.refl y) ≃ₜ GenLoop (N ⊕ Unit) Y y where
+  toFun := flattenPathLoop
+  invFun := curryPathLoop
+  left_inv := curry_flatten
+  right_inv := flatten_curry
+  continuous_toFun := continuous_flattenPathLoop
+  continuous_invFun := continuous_curryPathLoop
+
+def loopPiEquiv (y : Y) :
+    HomotopyGroup N (Path y y) (Path.refl y) ≃ HomotopyGroup (N ⊕ Unit) Y y :=
+  piEquivOfLoopHomeomorph loopSpaceCubeHomeomorph
+
+def pi9LoopEquivPi10 (y : Y) :
+    π_ 9 (Path y y) (Path.refl y) ≃ π_ 10 Y y :=
+  (loopPiEquiv y).trans (piReindexEquiv
+    ((Equiv.sumCongr (Equiv.refl (Fin 9)) (Equiv.ofUnique Unit (Fin 1))).trans finSumFinEquiv) y)
+
+end NearbyLagrangian.BasedHomotopy
+
+end
+
+end OAI

lean/OAI/Topology/Homotopy/OpenCollapse.lean

@@ -1,128 +1,128 @@
-import Mathlib
-
-namespace OAI
-
-/-! Continuous collapse to a one-point compactification and parameterized descent. -/
-
-noncomputable section
-universe u v w
-open Set ContinuousMap Topology
-open scoped Classical OnePoint Topology
-
-namespace NearbyLagrangian.OpenCollapse
-variable {Y : Type*}
-
-def collapse (U : Set Y) (y : Y) : OnePoint U :=
-  if h : y ∈ U then (⟨y,h⟩ : U) else ∞
-
-@[simp] lemma collapse_of_mem {U : Set Y} {y : Y} (h : y ∈ U) :
-    collapse U y = (⟨y,h⟩ : U) := by simp [collapse,h]
-@[simp] lemma collapse_of_notMem {U : Set Y} {y : Y} (h : y ∉ U) :
-    collapse U y = ∞ := by simp [collapse,h]
-lemma collapse_eq_infty {U : Set Y} {y : Y} : collapse U y = ∞ ↔ y ∉ U := by
-  by_cases h : y ∈ U <;> simp [h]
-lemma collapse_eq {U : Set Y} {y z : Y} :
-    collapse U y = collapse U z ↔ y = z ∨ y ∉ U ∧ z ∉ U := by
-  by_cases hy : y ∈ U
-  · by_cases hz : z ∈ U
-    · simp [collapse,hy,hz]
-    · have hn : y ≠ z := by rintro rfl; contradiction
-      simp [collapse,hy,hz,hn]
-  · by_cases hz : z ∈ U
-    · have hn : y ≠ z := by rintro rfl; contradiction
-      simp [collapse,hy,hz,hn]
-    · simp [collapse,hy,hz]
-
-variable [TopologicalSpace Y] [T2Space Y]
-
-lemma continuous_collapse {U : Set Y} (hU : IsOpen U) : Continuous (collapse U) := by
-  rw [continuous_def]
-  intro S hS
-  by_cases hi : ∞ ∈ S
-  · have hK := ((OnePoint.isOpen_iff_of_mem hi).mp hS).2
-    have he : (collapse U ⁻¹' S)ᶜ = Subtype.val '' (((↑) : U → OnePoint U) ⁻¹' S)ᶜ := by
-      ext y
-      by_cases hy : y ∈ U
-      · simp [hy,Set.mem_image,collapse]
-      · simp [hy,collapse,hi]
-    rw [← isClosed_compl_iff,he]
-    exact (hK.image continuous_subtype_val).isClosed
-  · have hV := (OnePoint.isOpen_iff_of_notMem hi).mp hS
-    have he : collapse U ⁻¹' S = Subtype.val '' (((↑) : U → OnePoint U) ⁻¹' S) := by
-      ext y
-      by_cases hy : y ∈ U
-      · simp [hy,Set.mem_image,collapse]
-      · simp [hy,collapse,hi]
-    rw [he]
-    exact hU.isOpenMap_subtype_val _ hV
-
-omit [TopologicalSpace Y] [T2Space Y] in
-lemma collapse_surjective {U : Set Y} (hU : Uᶜ.Nonempty) : Function.Surjective (collapse U) := by
-  intro z
-  induction z using OnePoint.rec with
-  | infty => obtain ⟨y,hy⟩ := hU; exact ⟨y,collapse_of_notMem hy⟩
-  | coe y => exact ⟨y,collapse_of_mem y.property⟩
-
-variable [CompactSpace Y]
-lemma isQuotientMap_collapse {U : Set Y} (hU : IsOpen U) (hUc : Uᶜ.Nonempty) :
-    IsQuotientMap (collapse U) := by
-  let : LocallyCompactSpace U := hU.locallyCompactSpace
-  exact (continuous_collapse hU).isClosedMap.isQuotientMap (continuous_collapse hU)
-    (collapse_surjective hUc)
-
-def descend {U : Set Y} (hU : IsOpen U) (hUc : Uᶜ.Nonempty)
-    {X : Type*} [TopologicalSpace X] (f : C(Y,X)) (x : X)
-    (hf : ∀ y ∉ U, f y = x) : C(OnePoint U,X) where
-  toFun := fun z => z.elim x (fun y => f y)
-  continuous_toFun := (isQuotientMap_collapse hU hUc).continuous_iff.mpr (by
-    have he : (fun z : OnePoint U => z.elim x (fun y => f y)) ∘ collapse U = f := by
-      funext y
-      by_cases hy : y ∈ U
-      · simp [hy]
-      · simp [hy,hf y hy]
-    rw [he]
-    exact f.continuous)
-
-@[simp] lemma descend_collapse {U : Set Y} (hU : IsOpen U) (hUc : Uᶜ.Nonempty)
-    {X : Type*} [TopologicalSpace X] (f : C(Y,X)) (x : X)
-    (hf : ∀ y ∉ U, f y = x) (y : Y) : descend hU hUc f x hf (collapse U y) = f y := by
-  by_cases hy : y ∈ U <;> simp [descend,hy,hf y]
-
-end NearbyLagrangian.OpenCollapse
-
-end
-
-noncomputable section
-universe u v w
-open Set ContinuousMap Topology
-open scoped Classical OnePoint Topology
-
-namespace NearbyLagrangian.OpenCollapse
-variable {Y P X : Type*} [TopologicalSpace Y] [T2Space Y] [CompactSpace Y]
-  [TopologicalSpace P] [T2Space P] [CompactSpace P] [TopologicalSpace X]
-
-lemma continuous_descend_param {U : Set Y} (hU : IsOpen U) (hUc : Uᶜ.Nonempty)
-    (F : C(P × Y,X)) (g : C(P,X)) (hF : ∀ p y, y ∉ U → F (p,y) = g p) :
-    Continuous (fun q : P × OnePoint U => q.2.elim (g q.1) (fun y => F (q.1,y))) := by
-  let : LocallyCompactSpace U := hU.locallyCompactSpace
-  let c : P × Y → P × OnePoint U := fun z => (z.1,collapse U z.2)
-  have hc : Continuous c := continuous_fst.prodMk ((continuous_collapse hU).comp continuous_snd)
-  have hs : Function.Surjective c := by
-    rintro ⟨p,z⟩
-    obtain ⟨y,hy⟩ := collapse_surjective hUc z
-    exact ⟨(p,y),Prod.ext rfl hy⟩
-  have hq := hc.isClosedMap.isQuotientMap hc hs
-  apply hq.continuous_iff.mpr
-  have he : (fun q : P × OnePoint U => q.2.elim (g q.1) (fun y => F (q.1,y))) ∘ c = F := by
-    funext q
-    rcases q with ⟨p,y⟩
-    change (collapse U y).elim (g p) (fun y => F (p,y)) = F (p,y)
-    by_cases hy : y ∈ U <;> simp [hy,hF p y]
-  rw [he]
-  exact F.continuous
-
-end NearbyLagrangian.OpenCollapse
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Continuous collapse to a one-point compactification and parameterized descent. -/
+
+noncomputable section
+universe u v w
+open Set ContinuousMap Topology
+open scoped Classical OnePoint Topology
+
+namespace NearbyLagrangian.OpenCollapse
+variable {Y : Type*}
+
+def collapse (U : Set Y) (y : Y) : OnePoint U :=
+  if h : y ∈ U then (⟨y,h⟩ : U) else ∞
+
+@[simp] lemma collapse_of_mem {U : Set Y} {y : Y} (h : y ∈ U) :
+    collapse U y = (⟨y,h⟩ : U) := by simp [collapse,h]
+@[simp] lemma collapse_of_notMem {U : Set Y} {y : Y} (h : y ∉ U) :
+    collapse U y = ∞ := by simp [collapse,h]
+lemma collapse_eq_infty {U : Set Y} {y : Y} : collapse U y = ∞ ↔ y ∉ U := by
+  by_cases h : y ∈ U <;> simp [h]
+lemma collapse_eq {U : Set Y} {y z : Y} :
+    collapse U y = collapse U z ↔ y = z ∨ y ∉ U ∧ z ∉ U := by
+  by_cases hy : y ∈ U
+  · by_cases hz : z ∈ U
+    · simp [collapse,hy,hz]
+    · have hn : y ≠ z := by rintro rfl; contradiction
+      simp [collapse,hy,hz,hn]
+  · by_cases hz : z ∈ U
+    · have hn : y ≠ z := by rintro rfl; contradiction
+      simp [collapse,hy,hz,hn]
+    · simp [collapse,hy,hz]
+
+variable [TopologicalSpace Y] [T2Space Y]
+
+lemma continuous_collapse {U : Set Y} (hU : IsOpen U) : Continuous (collapse U) := by
+  rw [continuous_def]
+  intro S hS
+  by_cases hi : ∞ ∈ S
+  · have hK := ((OnePoint.isOpen_iff_of_mem hi).mp hS).2
+    have he : (collapse U ⁻¹' S)ᶜ = Subtype.val '' (((↑) : U → OnePoint U) ⁻¹' S)ᶜ := by
+      ext y
+      by_cases hy : y ∈ U
+      · simp [hy,Set.mem_image,collapse]
+      · simp [hy,collapse,hi]
+    rw [← isClosed_compl_iff,he]
+    exact (hK.image continuous_subtype_val).isClosed
+  · have hV := (OnePoint.isOpen_iff_of_notMem hi).mp hS
+    have he : collapse U ⁻¹' S = Subtype.val '' (((↑) : U → OnePoint U) ⁻¹' S) := by
+      ext y
+      by_cases hy : y ∈ U
+      · simp [hy,Set.mem_image,collapse]
+      · simp [hy,collapse,hi]
+    rw [he]
+    exact hU.isOpenMap_subtype_val _ hV
+
+omit [TopologicalSpace Y] [T2Space Y] in
+lemma collapse_surjective {U : Set Y} (hU : Uᶜ.Nonempty) : Function.Surjective (collapse U) := by
+  intro z
+  induction z using OnePoint.rec with
+  | infty => obtain ⟨y,hy⟩ := hU; exact ⟨y,collapse_of_notMem hy⟩
+  | coe y => exact ⟨y,collapse_of_mem y.property⟩
+
+variable [CompactSpace Y]
+lemma isQuotientMap_collapse {U : Set Y} (hU : IsOpen U) (hUc : Uᶜ.Nonempty) :
+    IsQuotientMap (collapse U) := by
+  let : LocallyCompactSpace U := hU.locallyCompactSpace
+  exact (continuous_collapse hU).isClosedMap.isQuotientMap (continuous_collapse hU)
+    (collapse_surjective hUc)
+
+def descend {U : Set Y} (hU : IsOpen U) (hUc : Uᶜ.Nonempty)
+    {X : Type*} [TopologicalSpace X] (f : C(Y,X)) (x : X)
+    (hf : ∀ y ∉ U, f y = x) : C(OnePoint U,X) where
+  toFun := fun z => z.elim x (fun y => f y)
+  continuous_toFun := (isQuotientMap_collapse hU hUc).continuous_iff.mpr (by
+    have he : (fun z : OnePoint U => z.elim x (fun y => f y)) ∘ collapse U = f := by
+      funext y
+      by_cases hy : y ∈ U
+      · simp [hy]
+      · simp [hy,hf y hy]
+    rw [he]
+    exact f.continuous)
+
+@[simp] lemma descend_collapse {U : Set Y} (hU : IsOpen U) (hUc : Uᶜ.Nonempty)
+    {X : Type*} [TopologicalSpace X] (f : C(Y,X)) (x : X)
+    (hf : ∀ y ∉ U, f y = x) (y : Y) : descend hU hUc f x hf (collapse U y) = f y := by
+  by_cases hy : y ∈ U <;> simp [descend,hy,hf y]
+
+end NearbyLagrangian.OpenCollapse
+
+end
+
+noncomputable section
+universe u v w
+open Set ContinuousMap Topology
+open scoped Classical OnePoint Topology
+
+namespace NearbyLagrangian.OpenCollapse
+variable {Y P X : Type*} [TopologicalSpace Y] [T2Space Y] [CompactSpace Y]
+  [TopologicalSpace P] [T2Space P] [CompactSpace P] [TopologicalSpace X]
+
+lemma continuous_descend_param {U : Set Y} (hU : IsOpen U) (hUc : Uᶜ.Nonempty)
+    (F : C(P × Y,X)) (g : C(P,X)) (hF : ∀ p y, y ∉ U → F (p,y) = g p) :
+    Continuous (fun q : P × OnePoint U => q.2.elim (g q.1) (fun y => F (q.1,y))) := by
+  let : LocallyCompactSpace U := hU.locallyCompactSpace
+  let c : P × Y → P × OnePoint U := fun z => (z.1,collapse U z.2)
+  have hc : Continuous c := continuous_fst.prodMk ((continuous_collapse hU).comp continuous_snd)
+  have hs : Function.Surjective c := by
+    rintro ⟨p,z⟩
+    obtain ⟨y,hy⟩ := collapse_surjective hUc z
+    exact ⟨(p,y),Prod.ext rfl hy⟩
+  have hq := hc.isClosedMap.isQuotientMap hc hs
+  apply hq.continuous_iff.mpr
+  have he : (fun q : P × OnePoint U => q.2.elim (g q.1) (fun y => F (q.1,y))) ∘ c = F := by
+    funext q
+    rcases q with ⟨p,y⟩
+    change (collapse U y).elim (g p) (fun y => F (p,y)) = F (p,y)
+    by_cases hy : y ∈ U <;> simp [hy,hF p y]
+  rw [he]
+  exact F.continuous
+
+end NearbyLagrangian.OpenCollapse
+
+end
+
+end OAI

lean/OAI/Topology/Homotopy/RayTelescope.lean

@@ -1,383 +1,383 @@
-import Mathlib
-
-namespace OAI
-
-/-! Sequential ray telescopes, finite-stage representatives and path connectedness. -/
-
-noncomputable section
-universe u v w
-open Set ContinuousMap
-open scoped Topology
-
-namespace NearbyLagrangian.RayTelescope
-variable (X : ℕ → Type u) [∀ n, TopologicalSpace (X n)]
-variable (φ : ∀ m n, m ≤ n → C(X m,X n))
-
-abbrev Raw := Σ n, X n × {t : ℝ // (n:ℝ) ≤ t}
-
-inductive Glue : Raw X → Raw X → Prop
-  | step (n m : ℕ) (h : n ≤ m) (x : X n) (t : ℝ) (ht : (m:ℝ) ≤ t) :
-      Glue ⟨n,(x,⟨t,(by exact le_trans (by exact_mod_cast h) ht)⟩)⟩
-        ⟨m,(φ n m h x,⟨t,ht⟩)⟩
-
-abbrev Space := Quot (Glue X φ)
-
-def rawHeight : C(Raw X,ℝ) :=
-  ⟨fun a => a.2.2,continuous_sigma (fun _ => continuous_subtype_val.comp continuous_snd)⟩
-
-def height : C(Space X φ,ℝ) :=
-  ⟨Quot.lift (rawHeight X) (by intro a b h; cases h; rfl),
-   continuous_quot_lift _ (rawHeight X).continuous⟩
-
-@[simp] theorem height_quot (a : Raw X) : height X φ (Quot.mk _ a) = a.2.2 := rfl
-
-def bottomTime (n : ℕ) : {t : ℝ // (n:ℝ) ≤ t} := ⟨(n:ℝ),by exact le_rfl⟩
-
-def stage (n : ℕ) : C(X n,Space X φ) :=
-  ⟨fun x => Quot.mk _ ⟨n,(x,bottomTime n)⟩,by
-    exact continuous_quot_mk.comp (continuous_sigmaMk.comp
-      (continuous_id.prodMk (continuous_const :
-        Continuous (fun _ : X n => bottomTime n))))⟩
-
-@[simp] theorem height_stage (n : ℕ) (x : X n) : height X φ (stage X φ n x) = n := rfl
-
-variable [∀ n, Nonempty (X n)]
-variable [DirectedSystem X (fun {m n} h => (φ m n h : _ → _))]
-
-def rawCollapse (N : ℕ) : C(Raw X,X N) where
-  toFun a := if h : a.1 ≤ N then φ a.1 N h a.2.1 else Classical.choice inferInstance
-  continuous_toFun := continuous_sigma (fun n => by
-    by_cases h : n ≤ N
-    · simpa only [dite_eq_left h,Function.comp_def] using (φ n N h).continuous.comp continuous_fst
-    · simpa only [dite_eq_right h] using
-        (continuous_const : Continuous (fun _ : X n × {t : ℝ // (n:ℝ) ≤ t} => (Classical.choice inferInstance : X N))))
-
-omit [∀ n, Nonempty (X n)] in
-lemma stage_le_of_height_lt {N : ℕ} (a : Raw X) (h : rawHeight X a < (N:ℝ)+1) : a.1 ≤ N := by
-  have h' : (a.1:ℝ) < (N:ℝ)+1 := lt_of_le_of_lt a.2.2.2 h
-  have : a.1 < N+1 := by exact_mod_cast h'
-  omega
-
-lemma collapse_compatible {N : ℕ} {a b : Raw X}
-    (h : Quot.mk (Glue X φ) a = Quot.mk (Glue X φ) b)
-    (ha : rawHeight X a < (N:ℝ)+1) : rawCollapse X φ N a = rawCollapse X φ N b := by
-  have hrel := Quot.eqvGen_exact h
-  have aux : ∀ a b, Relation.EqvGen (Glue X φ) a b →
-      rawHeight X a = rawHeight X b ∧
-      (rawHeight X a < (N:ℝ)+1 → rawCollapse X φ N a = rawCollapse X φ N b) := by
-    intro a b h
-    induction h with
-    | rel a b hab =>
-      cases hab with
-      | step n m h x t ht =>
-        refine ⟨rfl,fun hz => ?_⟩
-        have hm : m ≤ N := stage_le_of_height_lt X ⟨m,(φ n m h x,⟨t,ht⟩)⟩ hz
-        have hn := h.trans hm
-        simp only [rawCollapse,ContinuousMap.coe_mk,dite_eq_left hn,dite_eq_left hm]
-        exact (DirectedSystem.map_map (f:=fun {m n} h => (φ m n h : _ → _)) h hm x).symm
-    | refl a => exact ⟨rfl,fun _ => rfl⟩
-    | symm a b hab ih => exact ⟨ih.1.symm,fun hb => (ih.2 (ih.1.symm ▸ hb)).symm⟩
-    | trans a b c hab hbc ihab ihbc =>
-      exact ⟨ihab.1.trans ihbc.1,fun ha => (ihab.2 ha).trans (ihbc.2 (ihab.1 ▸ ha))⟩
-  exact (aux a b hrel).2 ha
-
-abbrev Trunc (N : ℕ) := {z : Space X φ // height X φ z < (N:ℝ)+1}
-
-def representative (z : Space X φ) : Raw X := (Quot.mk_surjective z).choose
-
-omit [∀ n, Nonempty (X n)] [DirectedSystem X (fun {m n} h => (φ m n h : _ → _))] in
-@[simp] theorem representative_mk (z : Space X φ) :
-    Quot.mk (Glue X φ) (representative X φ z) = z := (Quot.mk_surjective z).choose_spec
-
-def retract (N : ℕ) : C(Trunc X φ N,X N) where
-  toFun z := rawCollapse X φ N (representative X φ z.1)
-  continuous_toFun := by
-    let s := {z : Space X φ | height X φ z < (N:ℝ)+1}
-    have hs : IsOpen s := isOpen_lt (height X φ).continuous continuous_const
-    have hq := (isQuotientMap_quot_mk (r:=Glue X φ)).restrictPreimage_isOpen hs
-    apply hq.continuous_iff.mpr
-    change Continuous (fun a : {a : Raw X // Quot.mk (Glue X φ) a ∈ s} =>
-      rawCollapse X φ N (representative X φ (Quot.mk _ a.1)))
-    have hf : (fun a : {a : Raw X // Quot.mk (Glue X φ) a ∈ s} =>
-      rawCollapse X φ N (representative X φ (Quot.mk _ a.1))) =
-        fun a => rawCollapse X φ N a.1 := by
-      funext a
-      apply collapse_compatible X φ
-      · exact representative_mk X φ _
-      · change height X φ (Quot.mk _ (representative X φ (Quot.mk _ a.1))) < _
-        rw [representative_mk]
-        exact a.2
-    rw [hf]
-    exact (rawCollapse X φ N).continuous.comp continuous_subtype_val
-
-@[simp] theorem retract_quot (N : ℕ) (a : Raw X)
-    (ha : rawHeight X a < (N:ℝ)+1) :
-    retract X φ N ⟨Quot.mk _ a,ha⟩ = rawCollapse X φ N a := by
-  apply collapse_compatible X φ
-  · exact representative_mk X φ _
-  · change height X φ (Quot.mk _ (representative X φ (Quot.mk _ a))) < _
-    rw [representative_mk]
-    exact ha
-
-@[simp] theorem retract_stage (N n : ℕ) (h : n ≤ N) (x : X n)
-    (hx : height X φ (stage X φ n x) < (N:ℝ)+1) :
-    retract X φ N ⟨stage X φ n x,hx⟩ = φ n N h x := by
-  calc
-    retract X φ N ⟨stage X φ n x,hx⟩ = rawCollapse X φ N ⟨n,(x,bottomTime n)⟩ :=
-      retract_quot X φ N ⟨n,(x,bottomTime n)⟩ hx
-    _ = φ n N h x := by simp only [rawCollapse,ContinuousMap.coe_mk,dite_eq_left h]
-
-end NearbyLagrangian.RayTelescope
-
-end
-
-noncomputable section
-universe u v w
-
-
-namespace NearbyLagrangian.RayTelescope
-open Set ContinuousMap
-open scoped Topology
-variable (X : ℕ → Type u) [∀ n, TopologicalSpace (X n)]
-variable (φ : ∀ m n, m ≤ n → C(X m,X n))
-
-def raiseTime (N : ℕ) (s : unitInterval) (t : ℝ) : ℝ := t + s * (max t N-t)
-
-lemma le_raiseTime (N : ℕ) (s : unitInterval) (t : ℝ) : t ≤ raiseTime N s t := by
-  unfold raiseTime
-  exact le_add_of_nonneg_right (mul_nonneg s.2.1 (sub_nonneg.mpr (le_max_left _ _)))
-
-@[simp] theorem raiseTime_zero (N : ℕ) (t : ℝ) : raiseTime N 0 t = t := by simp [raiseTime]
-@[simp] theorem raiseTime_one (N : ℕ) (t : ℝ) : raiseTime N 1 t = max t N := by simp [raiseTime]
-
-def cylinderRaise (N n : ℕ) : C((X n × {t : ℝ // (n:ℝ) ≤ t}) × unitInterval,Space X φ) where
-  toFun z := Quot.mk _ ⟨n,(z.1.1,⟨raiseTime N z.2 z.1.2,
-    z.1.2.2.trans (le_raiseTime _ _ _)⟩)⟩
-  continuous_toFun := by
-    have ht : Continuous (fun z : (X n × {t : ℝ // (n:ℝ) ≤ t}) × unitInterval => raiseTime N z.2 z.1.2) := by
-      unfold raiseTime
-      fun_prop
-    exact continuous_quot_mk.comp (continuous_sigmaMk.comp
-      (continuous_fst.fst.prodMk (ht.subtype_mk _)))
-
-def rawRaise (N : ℕ) : C(Raw X,C(unitInterval,Space X φ)) :=
-  ⟨fun a => (cylinderRaise X φ N a.1).curry a.2,
-   continuous_sigma (fun _ => (cylinderRaise X φ N _).curry.continuous)⟩
-
-def raiseFamily (N : ℕ) : C(Space X φ,C(unitInterval,Space X φ)) :=
-  ⟨Quot.lift (rawRaise X φ N) (by
-    intro a b h
-    ext s
-    cases h with
-    | step n m h x t ht =>
-      exact Quot.sound (Glue.step n m h x (raiseTime N s t) (ht.trans (le_raiseTime _ _ _)))),
-   continuous_quot_lift _ (rawRaise X φ N).continuous⟩
-
-@[simp] theorem raiseFamily_apply (N : ℕ) (a : Raw X) (s : unitInterval) :
-    raiseFamily X φ N (Quot.mk _ a) s = Quot.mk _
-      ⟨a.1,(a.2.1,⟨raiseTime N s a.2.2,a.2.2.2.trans (le_raiseTime _ _ _)⟩)⟩ := rfl
-
-def raised (N : ℕ) : C(Space X φ,Space X φ) :=
-  ⟨fun z => raiseFamily X φ N z 1,(continuous_eval_const (1:unitInterval)).comp
-    (raiseFamily X φ N).continuous⟩
-
-def raiseHomotopy (N : ℕ) : (ContinuousMap.id (Space X φ)).Homotopy (raised X φ N) where
-  toFun z := raiseFamily X φ N z.2 z.1
-  continuous_toFun := (raiseFamily X φ N).uncurry.continuous.comp continuous_swap
-  map_zero_left z := by
-    induction z using Quot.inductionOn with
-    | h a =>
-      change Quot.mk _ ⟨a.1,(a.2.1,⟨raiseTime N 0 a.2.2,_⟩)⟩ = Quot.mk _ a
-      exact congrArg (fun t : {t : ℝ // (a.1:ℝ) ≤ t} => Quot.mk (Glue X φ) ⟨a.1,(a.2.1,t)⟩)
-        (Subtype.ext (raiseTime_zero N a.2.2))
-  map_one_left _ := rfl
-
-@[simp] theorem height_raised (N : ℕ) (z : Space X φ) :
-    height X φ (raised X φ N z) = max (height X φ z) N := by
-  induction z using Quot.inductionOn with
-  | h a => change raiseTime N 1 a.2.2 = max (a.2.2:ℝ) N; exact raiseTime_one _ _
-
-variable [∀ n, Nonempty (X n)]
-variable [DirectedSystem X (fun {m n} h => (φ m n h : _ → _))]
-
-theorem raised_eq_stage_retract (N : ℕ) (z : Space X φ) (hz : height X φ z ≤ N) :
-    raised X φ N z = stage X φ N
-      (retract X φ N ⟨z,lt_of_le_of_lt hz (by linarith)⟩) := by
-  induction z using Quot.inductionOn with
-  | h a =>
-    have hn : a.1 ≤ N := stage_le_of_height_lt X a (lt_of_le_of_lt hz (by linarith))
-    have hr : retract X φ N ⟨Quot.mk _ a,lt_of_le_of_lt hz (by linarith)⟩ =
-        rawCollapse X φ N a := retract_quot X φ N a _
-    rw [hr]
-    change Quot.mk (Glue X φ) ⟨a.1,(a.2.1,⟨raiseTime N 1 a.2.2,_⟩)⟩ =
-      Quot.mk (Glue X φ) ⟨N,(rawCollapse X φ N a,⟨(N:ℝ),le_rfl⟩)⟩
-    have hz' : (a.2.2:ℝ) ≤ N := hz
-    simp only [raiseTime_one,max_eq_right hz',rawCollapse,ContinuousMap.coe_mk,dite_eq_left hn]
-    exact Quot.sound (Glue.step a.1 N hn a.2.1 (N:ℝ) le_rfl)
-
-end NearbyLagrangian.RayTelescope
-
-end
-
-noncomputable section
-universe u v w
-
-
-namespace NearbyLagrangian.RayTelescope
-open Set ContinuousMap
-open scoped Topology
-variable (X : ℕ → Type u) [∀ n, TopologicalSpace (X n)]
-variable (φ : ∀ m n, m ≤ n → C(X m,X n))
-variable {A : Type v} [TopologicalSpace A]
-
-theorem exists_height_bound [CompactSpace A] (f : C(A,Space X φ)) :
-    ∃ N : ℕ, ∀ a, height X φ (f a) ≤ N := by
-  obtain ⟨r,hr⟩ := (isCompact_range ((height X φ).continuous.comp f.continuous)).bddAbove
-  obtain ⟨N,hN⟩ := exists_nat_gt r
-  exact ⟨N,fun a => le_of_lt ((hr ⟨a,rfl⟩).trans_lt hN)⟩
-
-variable [∀ n, Nonempty (X n)]
-variable [DirectedSystem X (fun {m n} h => (φ m n h : _ → _))]
-
-def finiteMap (N : ℕ) (f : C(A,Space X φ)) (hf : ∀ a, height X φ (f a) < (N:ℝ)+1) : C(A,X N) :=
-  (retract X φ N).comp ⟨fun a => ⟨f a,hf a⟩,f.continuous.subtype_mk _⟩
-
-def finiteHomotopy (N : ℕ) (f : C(A,Space X φ)) (hf : ∀ a, height X φ (f a) ≤ N) :
-    f.Homotopy ((stage X φ N).comp
-      (finiteMap X φ N f (fun a => lt_of_le_of_lt (hf a) (by linarith)))) :=
-  ((raiseHomotopy X φ N).compContinuousMap f).cast (by ext; rfl) (by
-    ext a
-    exact raised_eq_stage_retract X φ N (f a) (hf a))
-
-theorem exists_finite_representative [CompactSpace A] (f : C(A,Space X φ)) :
-    ∃ N : ℕ, ∃ g : C(A,X N), f.Homotopic ((stage X φ N).comp g) := by
-  obtain ⟨N,hN⟩ := exists_height_bound X φ f
-  exact ⟨N,finiteMap X φ N f (fun a => lt_of_le_of_lt (hN a) (by linarith)),
-    ⟨finiteHomotopy X φ N f hN⟩⟩
-
-@[simp] theorem finiteMap_stage (N m : ℕ) (h : m ≤ N) (f : C(A,X m))
-    (hf : ∀ a, height X φ ((stage X φ m).comp f a) < (N:ℝ)+1) :
-    finiteMap X φ N ((stage X φ m).comp f) hf = (φ m N h).comp f := by
-  ext a
-  exact retract_stage X φ N m h (f a) (hf a)
-
-theorem stage_stabilize_homotopic (m N : ℕ) (h : m ≤ N) :
-    (stage X φ m).Homotopic ((stage X φ N).comp (φ m N h)) := by
-  have hh : ∀ x : X m, height X φ (stage X φ m x) ≤ N := by
-    intro x
-    change (m:ℝ) ≤ N
-    exact_mod_cast h
-  have H := finiteHomotopy X φ N (stage X φ m) hh
-  refine ⟨H.cast rfl ?_⟩
-  apply congrArg ((stage X φ N).comp)
-  ext x
-  exact retract_stage X φ N m h x _
-
-def finiteStageHomotopy {m n N : ℕ} (hm : m ≤ N) (hn : n ≤ N)
-    (f : C(A,X m)) (g : C(A,X n))
-    (H : ((stage X φ m).comp f).Homotopy ((stage X φ n).comp g))
-    (hH : ∀ z, height X φ (H z) < (N:ℝ)+1) :
-    ((φ m N hm).comp f).Homotopy ((φ n N hn).comp g) where
-  toFun z := retract X φ N ⟨H z,hH z⟩
-  continuous_toFun := (retract X φ N).continuous.comp (H.continuous.subtype_mk _)
-  map_zero_left a := by
-    have ha : height X φ (stage X φ m (f a)) < (N:ℝ)+1 := by
-      have := hH (0,a)
-      simpa only [ContinuousMap.Homotopy.apply_zero,ContinuousMap.comp_apply] using this
-    calc
-      retract X φ N ⟨H (0,a),hH (0,a)⟩ = retract X φ N ⟨stage X φ m (f a),ha⟩ :=
-        congrArg (retract X φ N) (Subtype.ext (H.map_zero_left a))
-      _ = _ := retract_stage X φ N m hm (f a) ha
-  map_one_left a := by
-    have ha : height X φ (stage X φ n (g a)) < (N:ℝ)+1 := by
-      have := hH (1,a)
-      simpa only [ContinuousMap.Homotopy.apply_one,ContinuousMap.comp_apply] using this
-    calc
-      retract X φ N ⟨H (1,a),hH (1,a)⟩ = retract X φ N ⟨stage X φ n (g a),ha⟩ :=
-        congrArg (retract X φ N) (Subtype.ext (H.map_one_left a))
-      _ = _ := retract_stage X φ N n hn (g a) ha
-
-theorem stage_homotopic_iff [CompactSpace A] {m n : ℕ} (f : C(A,X m)) (g : C(A,X n)) :
-    ((stage X φ m).comp f).Homotopic ((stage X φ n).comp g) ↔
-      ∃ (N : ℕ) (hm : m ≤ N) (hn : n ≤ N), ((φ m N hm).comp f).Homotopic ((φ n N hn).comp g) := by
-  constructor
-  · rintro ⟨H⟩
-    obtain ⟨K,hK⟩ := exists_height_bound X φ H.toContinuousMap
-    let N := max K (max m n)
-    have hm : m ≤ N := (le_max_left m n).trans (le_max_right _ _)
-    have hn : n ≤ N := (le_max_right m n).trans (le_max_right _ _)
-    have hKN : (K:ℝ) ≤ N := by exact_mod_cast (le_max_left K (max m n))
-    refine ⟨N,hm,hn,⟨finiteStageHomotopy X φ hm hn f g H ?_⟩⟩
-    intro z
-    exact lt_of_le_of_lt ((hK z).trans hKN) (by linarith)
-  · rintro ⟨N,hm,hn,h⟩
-    exact (((stage_stabilize_homotopic X φ m N hm).comp (ContinuousMap.Homotopic.refl f)).trans
-      ((ContinuousMap.Homotopic.refl (stage X φ N)).comp h)).trans
-      (((stage_stabilize_homotopic X φ n N hn).comp (ContinuousMap.Homotopic.refl g)).symm)
-
-theorem stage_nullhomotopic_iff [CompactSpace A] {m : ℕ} (f : C(A,X m)) :
-    ((stage X φ m).comp f).Nullhomotopic ↔
-      ∃ (N : ℕ) (hm : m ≤ N), ((φ m N hm).comp f).Nullhomotopic := by
-  constructor
-  · rintro ⟨c,⟨H⟩⟩
-    obtain ⟨K,hK⟩ := exists_height_bound X φ H.toContinuousMap
-    obtain ⟨J,hJ⟩ := exists_nat_gt (height X φ c)
-    let N := max K (max m J)
-    have hm : m ≤ N := (le_max_left m J).trans (le_max_right _ _)
-    have hKN : (K:ℝ) ≤ N := by exact_mod_cast (le_max_left K (max m J))
-    have hJN : (J:ℝ) ≤ N := by exact_mod_cast ((le_max_right m J).trans (le_max_right K (max m J)))
-    have hc : height X φ c < (N:ℝ)+1 := lt_of_lt_of_le hJ (by linarith)
-    have hH : ∀ z, height X φ (H z) < (N:ℝ)+1 :=
-      fun z => lt_of_le_of_lt ((hK z).trans hKN) (by linarith)
-    refine ⟨N,hm,retract X φ N ⟨c,hc⟩,⟨{
-      toFun := fun z => retract X φ N ⟨H z,hH z⟩
-      continuous_toFun := (retract X φ N).continuous.comp (H.continuous.subtype_mk _)
-      map_zero_left := ?_
-      map_one_left := ?_ }⟩⟩
-    · intro a
-      have ha : height X φ (stage X φ m (f a)) < (N:ℝ)+1 := by
-        simpa only [ContinuousMap.Homotopy.apply_zero,ContinuousMap.comp_apply] using hH (0,a)
-      calc
-        retract X φ N ⟨H (0,a),hH (0,a)⟩ = retract X φ N ⟨stage X φ m (f a),ha⟩ :=
-          congrArg (retract X φ N) (Subtype.ext (H.map_zero_left a))
-        _ = _ := retract_stage X φ N m hm (f a) ha
-    · intro a
-      exact congrArg (retract X φ N) (Subtype.ext (H.map_one_left a))
-  · rintro ⟨N,hm,c,h⟩
-    refine ⟨stage X φ N c,?_⟩
-    exact ((stage_stabilize_homotopic X φ m N hm).comp (ContinuousMap.Homotopic.refl f)).trans
-      ((ContinuousMap.Homotopic.refl (stage X φ N)).comp h)
-
-end NearbyLagrangian.RayTelescope
-
-end
-
-noncomputable section
-universe u v w
-
-
-namespace NearbyLagrangian.RayTelescope
-open ContinuousMap
-variable (X : ℕ → Type u) [∀ n, TopologicalSpace (X n)]
-variable (φ : ∀ m n, m ≤ n → C(X m,X n))
-variable [DirectedSystem X (fun {m n} h => (φ m n h : _ → _))]
-variable [∀ n, PathConnectedSpace (X n)]
-
-theorem joined_base (b : X 0) (z : Space X φ) : Joined z (stage X φ 0 b) := by
-  obtain ⟨N,g,⟨H⟩⟩ := exists_finite_representative X φ (ContinuousMap.const Unit z)
-  have h₁ : Joined z (stage X φ N (g ())) := ⟨H.evalAt ()⟩
-  have h₂ : Joined (stage X φ N (g ())) (stage X φ N (φ 0 N (Nat.zero_le N) b)) :=
-    (PathConnectedSpace.joined _ _).map (stage X φ N).continuous
-  obtain ⟨K⟩ := stage_stabilize_homotopic X φ 0 N (Nat.zero_le N)
-  exact (h₁.trans h₂).trans (Joined.symm ⟨K.evalAt b⟩)
-
-instance pathConnected : PathConnectedSpace (Space X φ) where
-  nonempty := ⟨stage X φ 0 (Classical.choice inferInstance)⟩
-  joined z w := (joined_base X φ (Classical.choice inferInstance) z).trans
-    (joined_base X φ (Classical.choice inferInstance) w).symm
-
-end NearbyLagrangian.RayTelescope
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Sequential ray telescopes, finite-stage representatives and path connectedness. -/
+
+noncomputable section
+universe u v w
+open Set ContinuousMap
+open scoped Topology
+
+namespace NearbyLagrangian.RayTelescope
+variable (X : ℕ → Type u) [∀ n, TopologicalSpace (X n)]
+variable (φ : ∀ m n, m ≤ n → C(X m,X n))
+
+abbrev Raw := Σ n, X n × {t : ℝ // (n:ℝ) ≤ t}
+
+inductive Glue : Raw X → Raw X → Prop
+  | step (n m : ℕ) (h : n ≤ m) (x : X n) (t : ℝ) (ht : (m:ℝ) ≤ t) :
+      Glue ⟨n,(x,⟨t,(by exact le_trans (by exact_mod_cast h) ht)⟩)⟩
+        ⟨m,(φ n m h x,⟨t,ht⟩)⟩
+
+abbrev Space := Quot (Glue X φ)
+
+def rawHeight : C(Raw X,ℝ) :=
+  ⟨fun a => a.2.2,continuous_sigma (fun _ => continuous_subtype_val.comp continuous_snd)⟩
+
+def height : C(Space X φ,ℝ) :=
+  ⟨Quot.lift (rawHeight X) (by intro a b h; cases h; rfl),
+   continuous_quot_lift _ (rawHeight X).continuous⟩
+
+@[simp] theorem height_quot (a : Raw X) : height X φ (Quot.mk _ a) = a.2.2 := rfl
+
+def bottomTime (n : ℕ) : {t : ℝ // (n:ℝ) ≤ t} := ⟨(n:ℝ),by exact le_rfl⟩
+
+def stage (n : ℕ) : C(X n,Space X φ) :=
+  ⟨fun x => Quot.mk _ ⟨n,(x,bottomTime n)⟩,by
+    exact continuous_quot_mk.comp (continuous_sigmaMk.comp
+      (continuous_id.prodMk (continuous_const :
+        Continuous (fun _ : X n => bottomTime n))))⟩
+
+@[simp] theorem height_stage (n : ℕ) (x : X n) : height X φ (stage X φ n x) = n := rfl
+
+variable [∀ n, Nonempty (X n)]
+variable [DirectedSystem X (fun {m n} h => (φ m n h : _ → _))]
+
+def rawCollapse (N : ℕ) : C(Raw X,X N) where
+  toFun a := if h : a.1 ≤ N then φ a.1 N h a.2.1 else Classical.choice inferInstance
+  continuous_toFun := continuous_sigma (fun n => by
+    by_cases h : n ≤ N
+    · simpa only [dite_eq_left h,Function.comp_def] using (φ n N h).continuous.comp continuous_fst
+    · simpa only [dite_eq_right h] using
+        (continuous_const : Continuous (fun _ : X n × {t : ℝ // (n:ℝ) ≤ t} => (Classical.choice inferInstance : X N))))
+
+omit [∀ n, Nonempty (X n)] in
+lemma stage_le_of_height_lt {N : ℕ} (a : Raw X) (h : rawHeight X a < (N:ℝ)+1) : a.1 ≤ N := by
+  have h' : (a.1:ℝ) < (N:ℝ)+1 := lt_of_le_of_lt a.2.2.2 h
+  have : a.1 < N+1 := by exact_mod_cast h'
+  omega
+
+lemma collapse_compatible {N : ℕ} {a b : Raw X}
+    (h : Quot.mk (Glue X φ) a = Quot.mk (Glue X φ) b)
+    (ha : rawHeight X a < (N:ℝ)+1) : rawCollapse X φ N a = rawCollapse X φ N b := by
+  have hrel := Quot.eqvGen_exact h
+  have aux : ∀ a b, Relation.EqvGen (Glue X φ) a b →
+      rawHeight X a = rawHeight X b ∧
+      (rawHeight X a < (N:ℝ)+1 → rawCollapse X φ N a = rawCollapse X φ N b) := by
+    intro a b h
+    induction h with
+    | rel a b hab =>
+      cases hab with
+      | step n m h x t ht =>
+        refine ⟨rfl,fun hz => ?_⟩
+        have hm : m ≤ N := stage_le_of_height_lt X ⟨m,(φ n m h x,⟨t,ht⟩)⟩ hz
+        have hn := h.trans hm
+        simp only [rawCollapse,ContinuousMap.coe_mk,dite_eq_left hn,dite_eq_left hm]
+        exact (DirectedSystem.map_map (f:=fun {m n} h => (φ m n h : _ → _)) h hm x).symm
+    | refl a => exact ⟨rfl,fun _ => rfl⟩
+    | symm a b hab ih => exact ⟨ih.1.symm,fun hb => (ih.2 (ih.1.symm ▸ hb)).symm⟩
+    | trans a b c hab hbc ihab ihbc =>
+      exact ⟨ihab.1.trans ihbc.1,fun ha => (ihab.2 ha).trans (ihbc.2 (ihab.1 ▸ ha))⟩
+  exact (aux a b hrel).2 ha
+
+abbrev Trunc (N : ℕ) := {z : Space X φ // height X φ z < (N:ℝ)+1}
+
+def representative (z : Space X φ) : Raw X := (Quot.mk_surjective z).choose
+
+omit [∀ n, Nonempty (X n)] [DirectedSystem X (fun {m n} h => (φ m n h : _ → _))] in
+@[simp] theorem representative_mk (z : Space X φ) :
+    Quot.mk (Glue X φ) (representative X φ z) = z := (Quot.mk_surjective z).choose_spec
+
+def retract (N : ℕ) : C(Trunc X φ N,X N) where
+  toFun z := rawCollapse X φ N (representative X φ z.1)
+  continuous_toFun := by
+    let s := {z : Space X φ | height X φ z < (N:ℝ)+1}
+    have hs : IsOpen s := isOpen_lt (height X φ).continuous continuous_const
+    have hq := (isQuotientMap_quot_mk (r:=Glue X φ)).restrictPreimage_isOpen hs
+    apply hq.continuous_iff.mpr
+    change Continuous (fun a : {a : Raw X // Quot.mk (Glue X φ) a ∈ s} =>
+      rawCollapse X φ N (representative X φ (Quot.mk _ a.1)))
+    have hf : (fun a : {a : Raw X // Quot.mk (Glue X φ) a ∈ s} =>
+      rawCollapse X φ N (representative X φ (Quot.mk _ a.1))) =
+        fun a => rawCollapse X φ N a.1 := by
+      funext a
+      apply collapse_compatible X φ
+      · exact representative_mk X φ _
+      · change height X φ (Quot.mk _ (representative X φ (Quot.mk _ a.1))) < _
+        rw [representative_mk]
+        exact a.2
+    rw [hf]
+    exact (rawCollapse X φ N).continuous.comp continuous_subtype_val
+
+@[simp] theorem retract_quot (N : ℕ) (a : Raw X)
+    (ha : rawHeight X a < (N:ℝ)+1) :
+    retract X φ N ⟨Quot.mk _ a,ha⟩ = rawCollapse X φ N a := by
+  apply collapse_compatible X φ
+  · exact representative_mk X φ _
+  · change height X φ (Quot.mk _ (representative X φ (Quot.mk _ a))) < _
+    rw [representative_mk]
+    exact ha
+
+@[simp] theorem retract_stage (N n : ℕ) (h : n ≤ N) (x : X n)
+    (hx : height X φ (stage X φ n x) < (N:ℝ)+1) :
+    retract X φ N ⟨stage X φ n x,hx⟩ = φ n N h x := by
+  calc
+    retract X φ N ⟨stage X φ n x,hx⟩ = rawCollapse X φ N ⟨n,(x,bottomTime n)⟩ :=
+      retract_quot X φ N ⟨n,(x,bottomTime n)⟩ hx
+    _ = φ n N h x := by simp only [rawCollapse,ContinuousMap.coe_mk,dite_eq_left h]
+
+end NearbyLagrangian.RayTelescope
+
+end
+
+noncomputable section
+universe u v w
+
+
+namespace NearbyLagrangian.RayTelescope
+open Set ContinuousMap
+open scoped Topology
+variable (X : ℕ → Type u) [∀ n, TopologicalSpace (X n)]
+variable (φ : ∀ m n, m ≤ n → C(X m,X n))
+
+def raiseTime (N : ℕ) (s : unitInterval) (t : ℝ) : ℝ := t + s * (max t N-t)
+
+lemma le_raiseTime (N : ℕ) (s : unitInterval) (t : ℝ) : t ≤ raiseTime N s t := by
+  unfold raiseTime
+  exact le_add_of_nonneg_right (mul_nonneg s.2.1 (sub_nonneg.mpr (le_max_left _ _)))
+
+@[simp] theorem raiseTime_zero (N : ℕ) (t : ℝ) : raiseTime N 0 t = t := by simp [raiseTime]
+@[simp] theorem raiseTime_one (N : ℕ) (t : ℝ) : raiseTime N 1 t = max t N := by simp [raiseTime]
+
+def cylinderRaise (N n : ℕ) : C((X n × {t : ℝ // (n:ℝ) ≤ t}) × unitInterval,Space X φ) where
+  toFun z := Quot.mk _ ⟨n,(z.1.1,⟨raiseTime N z.2 z.1.2,
+    z.1.2.2.trans (le_raiseTime _ _ _)⟩)⟩
+  continuous_toFun := by
+    have ht : Continuous (fun z : (X n × {t : ℝ // (n:ℝ) ≤ t}) × unitInterval => raiseTime N z.2 z.1.2) := by
+      unfold raiseTime
+      fun_prop
+    exact continuous_quot_mk.comp (continuous_sigmaMk.comp
+      (continuous_fst.fst.prodMk (ht.subtype_mk _)))
+
+def rawRaise (N : ℕ) : C(Raw X,C(unitInterval,Space X φ)) :=
+  ⟨fun a => (cylinderRaise X φ N a.1).curry a.2,
+   continuous_sigma (fun _ => (cylinderRaise X φ N _).curry.continuous)⟩
+
+def raiseFamily (N : ℕ) : C(Space X φ,C(unitInterval,Space X φ)) :=
+  ⟨Quot.lift (rawRaise X φ N) (by
+    intro a b h
+    ext s
+    cases h with
+    | step n m h x t ht =>
+      exact Quot.sound (Glue.step n m h x (raiseTime N s t) (ht.trans (le_raiseTime _ _ _)))),
+   continuous_quot_lift _ (rawRaise X φ N).continuous⟩
+
+@[simp] theorem raiseFamily_apply (N : ℕ) (a : Raw X) (s : unitInterval) :
+    raiseFamily X φ N (Quot.mk _ a) s = Quot.mk _
+      ⟨a.1,(a.2.1,⟨raiseTime N s a.2.2,a.2.2.2.trans (le_raiseTime _ _ _)⟩)⟩ := rfl
+
+def raised (N : ℕ) : C(Space X φ,Space X φ) :=
+  ⟨fun z => raiseFamily X φ N z 1,(continuous_eval_const (1:unitInterval)).comp
+    (raiseFamily X φ N).continuous⟩
+
+def raiseHomotopy (N : ℕ) : (ContinuousMap.id (Space X φ)).Homotopy (raised X φ N) where
+  toFun z := raiseFamily X φ N z.2 z.1
+  continuous_toFun := (raiseFamily X φ N).uncurry.continuous.comp continuous_swap
+  map_zero_left z := by
+    induction z using Quot.inductionOn with
+    | h a =>
+      change Quot.mk _ ⟨a.1,(a.2.1,⟨raiseTime N 0 a.2.2,_⟩)⟩ = Quot.mk _ a
+      exact congrArg (fun t : {t : ℝ // (a.1:ℝ) ≤ t} => Quot.mk (Glue X φ) ⟨a.1,(a.2.1,t)⟩)
+        (Subtype.ext (raiseTime_zero N a.2.2))
+  map_one_left _ := rfl
+
+@[simp] theorem height_raised (N : ℕ) (z : Space X φ) :
+    height X φ (raised X φ N z) = max (height X φ z) N := by
+  induction z using Quot.inductionOn with
+  | h a => change raiseTime N 1 a.2.2 = max (a.2.2:ℝ) N; exact raiseTime_one _ _
+
+variable [∀ n, Nonempty (X n)]
+variable [DirectedSystem X (fun {m n} h => (φ m n h : _ → _))]
+
+theorem raised_eq_stage_retract (N : ℕ) (z : Space X φ) (hz : height X φ z ≤ N) :
+    raised X φ N z = stage X φ N
+      (retract X φ N ⟨z,lt_of_le_of_lt hz (by linarith)⟩) := by
+  induction z using Quot.inductionOn with
+  | h a =>
+    have hn : a.1 ≤ N := stage_le_of_height_lt X a (lt_of_le_of_lt hz (by linarith))
+    have hr : retract X φ N ⟨Quot.mk _ a,lt_of_le_of_lt hz (by linarith)⟩ =
+        rawCollapse X φ N a := retract_quot X φ N a _
+    rw [hr]
+    change Quot.mk (Glue X φ) ⟨a.1,(a.2.1,⟨raiseTime N 1 a.2.2,_⟩)⟩ =
+      Quot.mk (Glue X φ) ⟨N,(rawCollapse X φ N a,⟨(N:ℝ),le_rfl⟩)⟩
+    have hz' : (a.2.2:ℝ) ≤ N := hz
+    simp only [raiseTime_one,max_eq_right hz',rawCollapse,ContinuousMap.coe_mk,dite_eq_left hn]
+    exact Quot.sound (Glue.step a.1 N hn a.2.1 (N:ℝ) le_rfl)
+
+end NearbyLagrangian.RayTelescope
+
+end
+
+noncomputable section
+universe u v w
+
+
+namespace NearbyLagrangian.RayTelescope
+open Set ContinuousMap
+open scoped Topology
+variable (X : ℕ → Type u) [∀ n, TopologicalSpace (X n)]
+variable (φ : ∀ m n, m ≤ n → C(X m,X n))
+variable {A : Type v} [TopologicalSpace A]
+
+theorem exists_height_bound [CompactSpace A] (f : C(A,Space X φ)) :
+    ∃ N : ℕ, ∀ a, height X φ (f a) ≤ N := by
+  obtain ⟨r,hr⟩ := (isCompact_range ((height X φ).continuous.comp f.continuous)).bddAbove
+  obtain ⟨N,hN⟩ := exists_nat_gt r
+  exact ⟨N,fun a => le_of_lt ((hr ⟨a,rfl⟩).trans_lt hN)⟩
+
+variable [∀ n, Nonempty (X n)]
+variable [DirectedSystem X (fun {m n} h => (φ m n h : _ → _))]
+
+def finiteMap (N : ℕ) (f : C(A,Space X φ)) (hf : ∀ a, height X φ (f a) < (N:ℝ)+1) : C(A,X N) :=
+  (retract X φ N).comp ⟨fun a => ⟨f a,hf a⟩,f.continuous.subtype_mk _⟩
+
+def finiteHomotopy (N : ℕ) (f : C(A,Space X φ)) (hf : ∀ a, height X φ (f a) ≤ N) :
+    f.Homotopy ((stage X φ N).comp
+      (finiteMap X φ N f (fun a => lt_of_le_of_lt (hf a) (by linarith)))) :=
+  ((raiseHomotopy X φ N).compContinuousMap f).cast (by ext; rfl) (by
+    ext a
+    exact raised_eq_stage_retract X φ N (f a) (hf a))
+
+theorem exists_finite_representative [CompactSpace A] (f : C(A,Space X φ)) :
+    ∃ N : ℕ, ∃ g : C(A,X N), f.Homotopic ((stage X φ N).comp g) := by
+  obtain ⟨N,hN⟩ := exists_height_bound X φ f
+  exact ⟨N,finiteMap X φ N f (fun a => lt_of_le_of_lt (hN a) (by linarith)),
+    ⟨finiteHomotopy X φ N f hN⟩⟩
+
+@[simp] theorem finiteMap_stage (N m : ℕ) (h : m ≤ N) (f : C(A,X m))
+    (hf : ∀ a, height X φ ((stage X φ m).comp f a) < (N:ℝ)+1) :
+    finiteMap X φ N ((stage X φ m).comp f) hf = (φ m N h).comp f := by
+  ext a
+  exact retract_stage X φ N m h (f a) (hf a)
+
+theorem stage_stabilize_homotopic (m N : ℕ) (h : m ≤ N) :
+    (stage X φ m).Homotopic ((stage X φ N).comp (φ m N h)) := by
+  have hh : ∀ x : X m, height X φ (stage X φ m x) ≤ N := by
+    intro x
+    change (m:ℝ) ≤ N
+    exact_mod_cast h
+  have H := finiteHomotopy X φ N (stage X φ m) hh
+  refine ⟨H.cast rfl ?_⟩
+  apply congrArg ((stage X φ N).comp)
+  ext x
+  exact retract_stage X φ N m h x _
+
+def finiteStageHomotopy {m n N : ℕ} (hm : m ≤ N) (hn : n ≤ N)
+    (f : C(A,X m)) (g : C(A,X n))
+    (H : ((stage X φ m).comp f).Homotopy ((stage X φ n).comp g))
+    (hH : ∀ z, height X φ (H z) < (N:ℝ)+1) :
+    ((φ m N hm).comp f).Homotopy ((φ n N hn).comp g) where
+  toFun z := retract X φ N ⟨H z,hH z⟩
+  continuous_toFun := (retract X φ N).continuous.comp (H.continuous.subtype_mk _)
+  map_zero_left a := by
+    have ha : height X φ (stage X φ m (f a)) < (N:ℝ)+1 := by
+      have := hH (0,a)
+      simpa only [ContinuousMap.Homotopy.apply_zero,ContinuousMap.comp_apply] using this
+    calc
+      retract X φ N ⟨H (0,a),hH (0,a)⟩ = retract X φ N ⟨stage X φ m (f a),ha⟩ :=
+        congrArg (retract X φ N) (Subtype.ext (H.map_zero_left a))
+      _ = _ := retract_stage X φ N m hm (f a) ha
+  map_one_left a := by
+    have ha : height X φ (stage X φ n (g a)) < (N:ℝ)+1 := by
+      have := hH (1,a)
+      simpa only [ContinuousMap.Homotopy.apply_one,ContinuousMap.comp_apply] using this
+    calc
+      retract X φ N ⟨H (1,a),hH (1,a)⟩ = retract X φ N ⟨stage X φ n (g a),ha⟩ :=
+        congrArg (retract X φ N) (Subtype.ext (H.map_one_left a))
+      _ = _ := retract_stage X φ N n hn (g a) ha
+
+theorem stage_homotopic_iff [CompactSpace A] {m n : ℕ} (f : C(A,X m)) (g : C(A,X n)) :
+    ((stage X φ m).comp f).Homotopic ((stage X φ n).comp g) ↔
+      ∃ (N : ℕ) (hm : m ≤ N) (hn : n ≤ N), ((φ m N hm).comp f).Homotopic ((φ n N hn).comp g) := by
+  constructor
+  · rintro ⟨H⟩
+    obtain ⟨K,hK⟩ := exists_height_bound X φ H.toContinuousMap
+    let N := max K (max m n)
+    have hm : m ≤ N := (le_max_left m n).trans (le_max_right _ _)
+    have hn : n ≤ N := (le_max_right m n).trans (le_max_right _ _)
+    have hKN : (K:ℝ) ≤ N := by exact_mod_cast (le_max_left K (max m n))
+    refine ⟨N,hm,hn,⟨finiteStageHomotopy X φ hm hn f g H ?_⟩⟩
+    intro z
+    exact lt_of_le_of_lt ((hK z).trans hKN) (by linarith)
+  · rintro ⟨N,hm,hn,h⟩
+    exact (((stage_stabilize_homotopic X φ m N hm).comp (ContinuousMap.Homotopic.refl f)).trans
+      ((ContinuousMap.Homotopic.refl (stage X φ N)).comp h)).trans
+      (((stage_stabilize_homotopic X φ n N hn).comp (ContinuousMap.Homotopic.refl g)).symm)
+
+theorem stage_nullhomotopic_iff [CompactSpace A] {m : ℕ} (f : C(A,X m)) :
+    ((stage X φ m).comp f).Nullhomotopic ↔
+      ∃ (N : ℕ) (hm : m ≤ N), ((φ m N hm).comp f).Nullhomotopic := by
+  constructor
+  · rintro ⟨c,⟨H⟩⟩
+    obtain ⟨K,hK⟩ := exists_height_bound X φ H.toContinuousMap
+    obtain ⟨J,hJ⟩ := exists_nat_gt (height X φ c)
+    let N := max K (max m J)
+    have hm : m ≤ N := (le_max_left m J).trans (le_max_right _ _)
+    have hKN : (K:ℝ) ≤ N := by exact_mod_cast (le_max_left K (max m J))
+    have hJN : (J:ℝ) ≤ N := by exact_mod_cast ((le_max_right m J).trans (le_max_right K (max m J)))
+    have hc : height X φ c < (N:ℝ)+1 := lt_of_lt_of_le hJ (by linarith)
+    have hH : ∀ z, height X φ (H z) < (N:ℝ)+1 :=
+      fun z => lt_of_le_of_lt ((hK z).trans hKN) (by linarith)
+    refine ⟨N,hm,retract X φ N ⟨c,hc⟩,⟨{
+      toFun := fun z => retract X φ N ⟨H z,hH z⟩
+      continuous_toFun := (retract X φ N).continuous.comp (H.continuous.subtype_mk _)
+      map_zero_left := ?_
+      map_one_left := ?_ }⟩⟩
+    · intro a
+      have ha : height X φ (stage X φ m (f a)) < (N:ℝ)+1 := by
+        simpa only [ContinuousMap.Homotopy.apply_zero,ContinuousMap.comp_apply] using hH (0,a)
+      calc
+        retract X φ N ⟨H (0,a),hH (0,a)⟩ = retract X φ N ⟨stage X φ m (f a),ha⟩ :=
+          congrArg (retract X φ N) (Subtype.ext (H.map_zero_left a))
+        _ = _ := retract_stage X φ N m hm (f a) ha
+    · intro a
+      exact congrArg (retract X φ N) (Subtype.ext (H.map_one_left a))
+  · rintro ⟨N,hm,c,h⟩
+    refine ⟨stage X φ N c,?_⟩
+    exact ((stage_stabilize_homotopic X φ m N hm).comp (ContinuousMap.Homotopic.refl f)).trans
+      ((ContinuousMap.Homotopic.refl (stage X φ N)).comp h)
+
+end NearbyLagrangian.RayTelescope
+
+end
+
+noncomputable section
+universe u v w
+
+
+namespace NearbyLagrangian.RayTelescope
+open ContinuousMap
+variable (X : ℕ → Type u) [∀ n, TopologicalSpace (X n)]
+variable (φ : ∀ m n, m ≤ n → C(X m,X n))
+variable [DirectedSystem X (fun {m n} h => (φ m n h : _ → _))]
+variable [∀ n, PathConnectedSpace (X n)]
+
+theorem joined_base (b : X 0) (z : Space X φ) : Joined z (stage X φ 0 b) := by
+  obtain ⟨N,g,⟨H⟩⟩ := exists_finite_representative X φ (ContinuousMap.const Unit z)
+  have h₁ : Joined z (stage X φ N (g ())) := ⟨H.evalAt ()⟩
+  have h₂ : Joined (stage X φ N (g ())) (stage X φ N (φ 0 N (Nat.zero_le N) b)) :=
+    (PathConnectedSpace.joined _ _).map (stage X φ N).continuous
+  obtain ⟨K⟩ := stage_stabilize_homotopic X φ 0 N (Nat.zero_le N)
+  exact (h₁.trans h₂).trans (Joined.symm ⟨K.evalAt b⟩)
+
+instance pathConnected : PathConnectedSpace (Space X φ) where
+  nonempty := ⟨stage X φ 0 (Classical.choice inferInstance)⟩
+  joined z w := (joined_base X φ (Classical.choice inferInstance) z).trans
+    (joined_base X φ (Classical.choice inferInstance) w).symm
+
+end NearbyLagrangian.RayTelescope
+
+end
+
+end OAI

lean/OAI/Topology/TorusCompression/Boundary.lean

@@ -1,572 +1,572 @@
-import Mathlib
-
-namespace OAI
-
-/-! Boundary restoration, bundle rank bounds and elementary torus compression. -/
-
-noncomputable section
-
-namespace TorusCompression
-
-open Set Metric
-open scoped Classical
-
-abbrev Slot (t : ℕ) := Fin t → ℝ
-abbrev Slots (t n : ℕ) := Fin n → Slot t
-
-def cubeSet (t n : ℕ) : Set (Slots t n) := {x | ∀ j, ‖x j‖ ≤ 1}
-abbrev Cube (t n : ℕ) := {x : Slots t n // x ∈ cubeSet t n}
-
-def OnBoundary {t : ℕ} (v : Slot t) : Prop := ‖v‖ = 1
-
-lemma slot_unit_ball (t : ℕ) :
-    {v : Slot t | ∀ i, v i ∈ Icc (-1 : ℝ) 1} = closedBall 0 1 := by
-  ext v
-  simp only [mem_ofPred_eq, mem_Icc, mem_closedBall, dist_zero_right]
-  rw [pi_norm_le_iff_of_nonneg (by norm_num : (0 : ℝ) ≤ 1)]
-  simp only [Real.norm_eq_abs, abs_le]
-
-lemma onBoundary_iff_frontier {t : ℕ} (v : Slot t) :
-    OnBoundary v ↔ v ∈ frontier {w : Slot t | ∀ i, w i ∈ Icc (-1 : ℝ) 1} := by
-  rw [slot_unit_ball, frontier_closedBall _ (by norm_num : (1 : ℝ) ≠ 0)]
-  simp [OnBoundary]
-
-def support {t n : ℕ} (x : Slots t n) : Finset (Fin n) :=
-  Finset.univ.filter (fun j => x j ≠ 0)
-
-def Sparse {t n : ℕ} (s : ℕ) : Set (Slots t n) := {x | (support x).card ≤ s}
-
-def sparseParameters (t n : ℕ) (s : ℝ) : Set (Slots t n) :=
-  {z | (∀ j, ‖z j‖ ≤ 2) ∧ ((support z).card : ℝ) ≤ s}
-
-def boundarySlots {t n : ℕ} (x : Slots t n) : Finset (Fin n) :=
-  Finset.univ.filter (fun j => OnBoundary (x j))
-
-@[simp] lemma mem_support {t n : ℕ} (x : Slots t n) (j : Fin n) :
-    j ∈ support x ↔ x j ≠ 0 := by simp [support]
-
-@[simp] lemma mem_boundarySlots {t n : ℕ} (x : Slots t n) (j : Fin n) :
-    j ∈ boundarySlots x ↔ OnBoundary (x j) := by simp [boundarySlots]
-
-@[simp] lemma support_zero (t n : ℕ) : support (0 : Slots t n) = ∅ := by
-  ext j
-  simp
-
-lemma sparse_nonempty (t n s : ℕ) : (Sparse (t := t) (n := n) s).Nonempty := by
-  exact ⟨0, by simp [Sparse]⟩
-
-lemma sparse_eq_union (t n s : ℕ) :
-    Sparse (t := t) (n := n) s =
-      ⋃ A : {A : Finset (Fin n) // A.card ≤ s},
-        {x : Slots t n | ∀ j, j ∉ A.val → x j = 0} := by
-  ext x
-  constructor
-  · intro hx
-    refine mem_iUnion.mpr ⟨⟨support x, hx⟩, ?_⟩
-    intro j hj
-    simpa using hj
-  · intro hx
-    obtain ⟨A, hA⟩ := mem_iUnion.mp hx
-    exact (Finset.card_le_card (show support x ⊆ A.val from by
-      intro j hj
-      by_contra hjA
-      exact (mem_support x j).mp hj (hA j hjA))).trans A.property
-
-lemma isClosed_sparse (t n s : ℕ) : IsClosed (Sparse (t := t) (n := n) s) := by
-  rw [sparse_eq_union]
-  apply isClosed_iUnion_of_finite
-  intro A
-  simp only [ofPred_forall]
-  exact isClosed_iInter fun j => isClosed_iInter fun _ =>
-    isClosed_eq (continuous_apply j)
-      (continuous_const : Continuous (fun _ : Slots t n => (0 : Slot t)))
-
-lemma sparseParameters_eq_empty {t n : ℕ} {s : ℝ} (hs : s < 0) :
-    sparseParameters t n s = ∅ := by
-  apply Set.eq_empty_iff_forall_notMem.mpr
-  intro x hx
-  have hcard : (0 : ℝ) ≤ (support x).card := Nat.cast_nonneg _
-  exact (not_le_of_gt hs) (hcard.trans hx.2)
-
-lemma sparseParameters_eq_inter {t n : ℕ} {s : ℝ} (hs : 0 ≤ s) :
-    sparseParameters t n s = closedBall (0 : Slots t n) 2 ∩ Sparse ⌊s⌋₊ := by
-  ext x
-  simp only [sparseParameters, Sparse, mem_ofPred_eq, mem_inter_iff,
-    mem_closedBall, dist_zero_right]
-  rw [pi_norm_le_iff_of_nonneg (by norm_num : (0 : ℝ) ≤ 2)]
-  exact and_congr Iff.rfl (Nat.le_floor_iff hs).symm
-
-lemma isCompact_sparseParameters (t n : ℕ) (s : ℝ) :
-    IsCompact (sparseParameters t n s) := by
-  by_cases hs : 0 ≤ s
-  · rw [sparseParameters_eq_inter hs]
-    exact (isCompact_closedBall (0 : Slots t n) 2).inter_right (isClosed_sparse t n ⌊s⌋₊)
-  · rw [sparseParameters_eq_empty (lt_of_not_ge hs)]
-    exact isCompact_empty
-
-lemma sparseParameters_eq_parameterCube {t n : ℕ} {s : ℝ} (hs : (n : ℝ) ≤ s) :
-    sparseParameters t n s = {z : Slots t n | ∀ j, ‖z j‖ ≤ 2} := by
-  ext z
-  refine ⟨fun hz => hz.1, fun hz => ⟨hz, ?_⟩⟩
-  have hcard : (support z).card ≤ n := by
-    exact (Finset.card_le_card (Finset.filter_subset _ _)).trans_eq (Finset.card_fin n)
-  exact (Nat.cast_le.mpr hcard).trans hs
-
-def restorationRadius {t n : ℕ} (s : ℕ) (x : Slots t n) : ℝ :=
-  min (1 / 2) (infDist x (Sparse s) / 2)
-
-lemma continuous_restorationRadius (t n s : ℕ) :
-    Continuous (restorationRadius (t := t) (n := n) s) := by
-  unfold restorationRadius
-  exact continuous_const.min ((continuous_infDist_pt (Sparse s)).div_const 2)
-
-lemma restorationRadius_pos {t n s : ℕ} {x : Slots t n} (hx : x ∉ Sparse s) :
-    0 < restorationRadius s x := by
-  have h := (isClosed_sparse t n s).notMem_iff_infDist_pos (sparse_nonempty t n s) |>.mp hx
-  exact lt_min (by norm_num) (by positivity)
-
-lemma restorationRadius_le_half {t n s : ℕ} (x : Slots t n) :
-    restorationRadius s x ≤ 1 / 2 := min_le_left _ _
-
-lemma restorationRadius_lt_infDist {t n s : ℕ} {x : Slots t n}
-    (hx : x ∉ Sparse s) : restorationRadius s x < infDist x (Sparse s) := by
-  have h := (isClosed_sparse t n s).notMem_iff_infDist_pos (sparse_nonempty t n s) |>.mp hx
-  exact (min_le_right _ _).trans_lt (by linarith)
-
-lemma many_large_slots {t n s : ℕ} {x : Slots t n} (hx : x ∉ Sparse s) :
-    s < (Finset.univ.filter (fun j => restorationRadius s x ≤ ‖x j‖)).card := by
-  let δ := restorationRadius s x
-  have hδ : 0 < δ := restorationRadius_pos hx
-  let A : Finset (Fin n) := Finset.univ.filter (fun j => δ ≤ ‖x j‖)
-  let y : Slots t n := fun j => if j ∈ A then x j else 0
-  by_contra h
-  have hA : A.card ≤ s := Nat.le_of_not_lt h
-  have hy : y ∈ Sparse s := by
-    apply le_trans (Finset.card_le_card (show support y ⊆ A from ?_)) hA
-    intro j hj
-    by_contra hjA
-    have hnonzero := (mem_support y j).mp hj
-    exact hnonzero (by simp [y, hjA])
-  have hdist : dist x y ≤ δ := by
-    rw [dist_eq_norm, pi_norm_le_iff_of_nonneg hδ.le]
-    intro j
-    by_cases hj : j ∈ A
-    · simp [y, hj, hδ.le]
-    · have hj' : ‖x j‖ < δ := by simpa [A] using hj
-      simpa [y, hj] using hj'.le
-  have hlt := restorationRadius_lt_infDist hx
-  exact not_lt_of_ge ((infDist_le_dist_of_mem hy).trans hdist) hlt
-
-def radial {t : ℕ} (δ : ℝ) (v : Slot t) : Slot t :=
-  (max δ ‖v‖)⁻¹ • v
-
-lemma norm_radial_le_one {t : ℕ} {δ : ℝ} (hδ : 0 < δ) (v : Slot t) :
-    ‖radial δ v‖ ≤ 1 := by
-  have hden : 0 < max δ ‖v‖ := hδ.trans_le (le_max_left _ _)
-  rw [radial, norm_smul, Real.norm_eq_abs, abs_inv,
-    abs_of_pos hden, ← div_eq_inv_mul]
-  exact (div_le_one hden).mpr (le_max_right _ _)
-
-lemma radial_boundary {t : ℕ} {δ : ℝ} (hδ : 0 < δ) {v : Slot t}
-    (hv : δ ≤ ‖v‖) : OnBoundary (radial δ v) := by
-  have hn : 0 < ‖v‖ := hδ.trans_le hv
-  simp [OnBoundary, radial, max_eq_right hv, norm_smul, inv_mul_cancel₀ hn.ne']
-
-lemma radial_fixes_boundary {t : ℕ} {δ : ℝ} (hδ : δ ≤ 1) {v : Slot t}
-    (hv : OnBoundary v) : radial δ v = v := by
-  simp [radial, show ‖v‖ = 1 from hv, max_eq_right hδ]
-
-def cutoff {t : ℕ} (v : Slot t) : ℝ := max 0 (4 * ‖v‖ - 3)
-
-def blend {t : ℕ} (v w : Slot t) : Slot t :=
-  (1 - cutoff v) • w + cutoff v • v
-
-lemma cutoff_range {t : ℕ} {v : Slot t} (hv : ‖v‖ ≤ 1) :
-    0 ≤ cutoff v ∧ cutoff v ≤ 1 := by
-  refine ⟨le_max_left _ _, max_le (by norm_num) ?_⟩
-  linarith
-
-lemma blend_sub {t : ℕ} (v w : Slot t) :
-    blend v w - v = (1 - cutoff v) • (w - v) := by
-  simp only [blend, smul_sub, sub_smul, one_smul]
-  abel
-
-lemma norm_blend_sub_le {t : ℕ} {v w : Slot t} (hv : ‖v‖ ≤ 1) :
-    ‖blend v w - v‖ ≤ ‖w - v‖ := by
-  obtain ⟨h0, h1⟩ := cutoff_range hv
-  rw [blend_sub, norm_smul, Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr h1)]
-  exact mul_le_of_le_one_left (norm_nonneg _) (by linarith)
-
-lemma blend_nonzero {t : ℕ} {v w : Slot t} (hv : ‖v‖ ≤ 1)
-    (hvw : ‖w - v‖ < 1 / 2) (hw : w ≠ 0) : blend v w ≠ 0 := by
-  by_cases hχ : cutoff v = 0
-  · simpa [blend, hχ] using hw
-  · have hχpos : 0 < cutoff v := lt_of_le_of_ne (cutoff_range hv).1 (Ne.symm hχ)
-    have hnorm : 3 / 4 < ‖v‖ := by
-      have h := (lt_max_iff.mp hχpos).resolve_left (lt_irrefl 0)
-      linarith
-    intro hzero
-    have h := (norm_blend_sub_le hv).trans_lt hvw
-    rw [hzero, zero_sub, norm_neg] at h
-    linarith
-
-lemma blend_fixes_boundary {t : ℕ} {v w : Slot t} (hv : OnBoundary v) :
-    blend v w = v := by
-  have hχ : cutoff v = 1 := by simp [cutoff, show ‖v‖ = 1 from hv]; norm_num
-  simp [blend, hχ]
-
-lemma continuous_blend {t : ℕ} : Continuous (fun p : Slot t × Slot t => blend p.1 p.2) := by
-  unfold blend cutoff
-  fun_prop
-
-lemma continuous_blend_of {X : Type*} [TopologicalSpace X] {t : ℕ}
-    {v w : X → Slot t} (hv : Continuous v) (hw : Continuous w) :
-    Continuous (fun x => blend (v x) (w x)) := by
-  have hc : Continuous (fun x => cutoff (v x)) :=
-    continuous_const.max ((continuous_const.mul hv.norm).sub continuous_const)
-  exact ((continuous_const.sub hc).smul hw).add (hc.smul hv)
-
-theorem boundary_restoration {X : Type*} [TopologicalSpace X]
-    {t n s : ℕ} (f : X → Cube t n) (g : X → Slots t n)
-    (hf : Continuous f) (hg : Continuous g)
-    (havoid : ∀ x, g x ∉ Sparse s)
-    (hclose : ∀ x, ‖g x - (f x).val‖ < 1 / 2) :
-    ∃ G : X → Cube t n, Continuous G ∧
-      (∀ x, s + 1 ≤ (boundarySlots (G x).val).card) ∧
-      (∀ x j, OnBoundary ((f x).val j) → (G x).val j = (f x).val j) := by
-  let g' : X → Slots t n := fun x j => blend ((f x).val j) (g x j)
-  have hfj : ∀ j, Continuous (fun x => (f x).val j) := fun j =>
-    (continuous_apply j).comp hf.subtype_val
-  have hgj : ∀ j, Continuous (fun x => g x j) := fun j =>
-    (continuous_apply j).comp hg
-  have hg' : Continuous g' := by
-    apply continuous_pi
-    intro j
-    exact continuous_blend_of (hfj j) (hgj j)
-  have havoid' : ∀ x, g' x ∉ Sparse s := by
-    intro x hx
-    apply havoid x
-    apply le_trans (Finset.card_le_card (show support (g x) ⊆ support (g' x) from ?_)) hx
-    intro j hj
-    apply (mem_support _ _).mpr
-    apply blend_nonzero ((f x).property j) _ ((mem_support _ _).mp hj)
-    exact (norm_le_pi_norm (g x - (f x).val) j).trans_lt (hclose x)
-  let G : X → Cube t n := fun x =>
-    ⟨fun j => radial (restorationRadius s (g' x)) (g' x j),
-      fun j => norm_radial_le_one (restorationRadius_pos (havoid' x)) (g' x j)⟩
-  refine ⟨G, ?_, ?_, ?_⟩
-  · apply Continuous.subtype_mk
-    apply continuous_pi
-    intro j
-    have hden : ∀ x, max (restorationRadius s (g' x)) ‖g' x j‖ ≠ 0 := by
-      intro x
-      exact ne_of_gt ((restorationRadius_pos (havoid' x)).trans_le (le_max_left _ _))
-    exact (((continuous_restorationRadius t n s).comp hg').max
-      (((continuous_apply j).comp hg').norm)).inv₀ hden |>.smul
-        ((continuous_apply j).comp hg')
-  · intro x
-    apply (Nat.succ_le_of_lt (many_large_slots (havoid' x))).trans
-    apply Finset.card_le_card
-    intro j hj
-    exact (mem_boundarySlots _ _).mpr
-      (radial_boundary (restorationRadius_pos (havoid' x)) ((Finset.mem_filter.mp hj).2))
-  · intro x j hj
-    change radial (restorationRadius s (g' x)) (blend ((f x).val j) (g x j)) = _
-    rw [blend_fixes_boundary hj]
-    exact radial_fixes_boundary ((restorationRadius_le_half (g' x)).trans (by norm_num)) hj
-
-theorem boundary_restoration_of_sparseParameters {X : Type*} [TopologicalSpace X]
-    {t n s : ℕ} (f : X → Cube t n) (g : X → Slots t n)
-    (hf : Continuous f) (hg : Continuous g)
-    (havoid : ∀ x, g x ∉ sparseParameters t n (s : ℝ))
-    (hclose : ∀ x, ‖g x - (f x).val‖ < 1 / 2) :
-    ∃ G : X → Cube t n, Continuous G ∧
-      (∀ x, s + 1 ≤ (boundarySlots (G x).val).card) ∧
-      (∀ x j, OnBoundary ((f x).val j) → (G x).val j = (f x).val j) := by
-  apply boundary_restoration f g hf hg ?_ hclose
-  intro x hx
-  apply havoid x
-  refine ⟨?_, ?_⟩
-  · intro j
-    have hj := (norm_le_pi_norm (g x - (f x).val) j).trans_lt (hclose x)
-    have hfbound := (f x).property j
-    have hadd := norm_le_norm_sub_add (g x j) ((f x).val j)
-    dsimp only [Pi.sub_apply] at hj
-    linarith
-  · exact_mod_cast hx
-
-def torusLocus {Y : Type*} {t n : ℕ} (f : Y → Cube t n)
-    (L : Set (Slot t)) (Z : Set (Slots t n)) : Set (Y × Z) :=
-  {q | ∀ j, (f q.1).val j - q.2.val j ∈ L}
-
-abbrev TorusLocus {Y : Type*} {t n : ℕ} (f : Y → Cube t n)
-    (L : Set (Slot t)) (Z : Set (Slots t n)) := ↥(torusLocus f L Z)
-
-def locusDifference {Y : Type*} {t n : ℕ} {f : Y → Cube t n}
-    {L : Set (Slot t)} {Z : Set (Slots t n)} (q : TorusLocus f L Z) : Fin n → L :=
-  fun j => ⟨(f q.val.1).val j - q.val.2.val j, q.property j⟩
-
-lemma isClosed_torusLocus {Y : Type*} [TopologicalSpace Y] {t n : ℕ}
-    {f : Y → Cube t n} {L : Set (Slot t)} {Z : Set (Slots t n)}
-    (hf : Continuous f) (hL : IsClosed L) : IsClosed (torusLocus f L Z) := by
-  unfold torusLocus
-  simp only [ofPred_forall]
-  apply isClosed_iInter
-  intro j
-  apply hL.preimage
-  have hc₁ : Continuous (fun q : Y × Z => (f q.1).val j) :=
-    (continuous_apply j).comp (hf.subtype_val.comp continuous_fst)
-  have hc₂ : Continuous (fun q : Y × Z => q.2.val j) :=
-    (continuous_apply j).comp (continuous_subtype_val.comp continuous_snd)
-  exact hc₁.sub hc₂
-
-lemma isCompact_sparse_torusLocus {Y : Type*} [TopologicalSpace Y] [CompactSpace Y]
-    {t n : ℕ} {f : Y → Cube t n} {L : Set (Slot t)}
-    (hf : Continuous f) (hL : IsClosed L) (s : ℝ) :
-    IsCompact (torusLocus f L (sparseParameters t n s)) := by
-  let : CompactSpace (sparseParameters t n s) :=
-    isCompact_iff_compactSpace.mp (isCompact_sparseParameters t n s)
-  exact (isClosed_torusLocus hf hL).isCompact
-
-lemma continuous_locusDifference {Y : Type*} [TopologicalSpace Y] {t n : ℕ}
-    {f : Y → Cube t n} {L : Set (Slot t)} {Z : Set (Slots t n)}
-    (hf : Continuous f) : Continuous (locusDifference (f := f) (L := L) (Z := Z)) := by
-  apply continuous_pi
-  intro j
-  apply Continuous.subtype_mk
-  have hc₁ : Continuous (fun q : TorusLocus f L Z => (f q.val.1).val j) :=
-    (continuous_apply j).comp
-      (hf.subtype_val.comp (continuous_fst.comp continuous_subtype_val))
-  have hc₂ : Continuous (fun q : TorusLocus f L Z => q.val.2.val j) :=
-    (continuous_apply j).comp (continuous_subtype_val.comp
-      (continuous_snd.comp continuous_subtype_val))
-  exact hc₁.sub hc₂
-
-def slotBundleFiber {Y : Type*} {t n r : ℕ} {f : Y → Cube t n}
-    {L : Set (Slot t)} {Z : Set (Slots t n)}
-    (H : L → Submodule ℂ (Fin r → ���)) (q : TorusLocus f L Z) :
-    Submodule ℂ (Fin n → Fin r → ℂ) :=
-  Submodule.pi univ (fun j => H (locusDifference q j))
-
-@[simp] lemma mem_slotBundleFiber {Y : Type*} {t n r : ℕ} {f : Y → Cube t n}
-    {L : Set (Slot t)} {Z : Set (Slots t n)} (H : L → Submodule ℂ (Fin r → ℂ))
-    (q : TorusLocus f L Z) (v : Fin n → Fin r → ℂ) :
-    v ∈ slotBundleFiber H q ↔ ∀ j, v j ∈ H (locusDifference q j) := by
-  simp [slotBundleFiber, Submodule.mem_pi]
-
-def EmbeddedTotal {B V : Type*} [AddCommGroup V] [Module ℂ V]
-    (E : B → Submodule ℂ V) := {q : B × V // q.2 ∈ E q.1}
-
-instance {B V : Type*} [TopologicalSpace B] [TopologicalSpace V]
-    [AddCommGroup V] [Module ℂ V] (E : B → Submodule ℂ V) :
-    TopologicalSpace (EmbeddedTotal E) := inferInstanceAs (TopologicalSpace {q : B × V // q.2 ∈ E q.1})
-
-structure BundleMono {B V : Type*} [TopologicalSpace B] [TopologicalSpace V]
-    [AddCommGroup V] [Module ℂ V] (E : B → Submodule ℂ V) (K : ℕ) where
-  map : ∀ b, E b →ₗ[ℂ] (Fin K → ℂ)
-  injective : ∀ b, Function.Injective (map b)
-  continuous_total : Continuous (fun q : EmbeddedTotal E => map q.val.1 ⟨q.val.2, q.property⟩)
-
-def slotBundleFiberEquiv {Y : Type*} {t n r : ℕ} {f : Y → Cube t n}
-    {L : Set (Slot t)} {Z : Set (Slots t n)}
-    (H : L → Submodule ℂ (Fin r → ℂ)) (q : TorusLocus f L Z) :
-    slotBundleFiber H q ≃ₗ[ℂ] ((j : Fin n) → H (locusDifference q j)) where
-  toFun v j := ⟨v.val j, (mem_slotBundleFiber H q v.val).mp v.property j⟩
-  invFun v := ⟨fun j => (v j).val,
-    (mem_slotBundleFiber H q _).mpr (fun j => (v j).property)⟩
-  left_inv _ := rfl
-  right_inv _ := rfl
-  map_add' _ _ := rfl
-  map_smul' _ _ := rfl
-
-lemma finrank_slotBundleFiber {Y : Type*} {t n r p : ℕ} {f : Y → Cube t n}
-    {L : Set (Slot t)} {Z : Set (Slots t n)}
-    (H : L → Submodule ℂ (Fin r → ℂ)) (hH : ∀ l, Module.finrank ℂ (H l) = p)
-    (q : TorusLocus f L Z) : Module.finrank ℂ (slotBundleFiber H q) = p * n := by
-  rw [(slotBundleFiberEquiv H q).finrank_eq, Module.finrank_pi_fintype]
-  simp [hH, Nat.mul_comm]
-
-lemma BundleMono.finrank_le {B V : Type*} [TopologicalSpace B] [TopologicalSpace V]
-    [AddCommGroup V] [Module ℂ V] {E : B → Submodule ℂ V} {K : ℕ}
-    (M : BundleMono E K) (b : B) : Module.finrank ℂ (E b) ≤ K := by
-  have h := LinearMap.finrank_le_finrank_of_injective (M.injective b)
-  simpa using h
-
-theorem rank_le_target_of_nonempty_locus {Y : Type*} [TopologicalSpace Y]
-    {t n r p K : ℕ} {f : Y → Cube t n}
-    {L : Set (Slot t)} {Z : Set (Slots t n)}
-    (H : L → Submodule ℂ (Fin r → ℂ)) (hH : ∀ l, Module.finrank ℂ (H l) = p)
-    (M : BundleMono (slotBundleFiber H (f := f) (Z := Z)) K)
-    (hq : (torusLocus f L Z).Nonempty) : p * n ≤ K := by
-  obtain ⟨q, hq⟩ := hq
-  simpa [finrank_slotBundleFiber H hH] using M.finrank_le (⟨q, hq⟩ : TorusLocus f L Z)
-
-def slotIndex {u n : ℕ} (b : Fin u) (j : Fin n) : Fin (u * n) :=
-  finProdFinEquiv (b, j)
-
-@[simp] lemma slotIndex_val {u n : ℕ} (b : Fin u) (j : Fin n) :
-    (slotIndex b j).val = b.val * n + j.val := by
-  simp [slotIndex, finProdFinEquiv, Nat.mul_comm, Nat.add_comm]
-
-def powerMap {X Y : Type*} (q : X → Y) (u : ℕ) (x : Fin u → X) : Fin u → Y :=
-  fun b => q (x b)
-
-lemma continuous_powerMap {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
-    {q : X → Y} (hq : Continuous q) (u : ℕ) : Continuous (powerMap q u) := by
-  exact continuous_pi fun b => hq.comp (continuous_apply b)
-
-def concatenate {X : Type*} {t n : ℕ} (f : X → Cube t n) (u : ℕ)
-    (x : Fin u → X) : Cube t (u * n) :=
-  ⟨fun k => (f (x (finProdFinEquiv.symm k).1)).val (finProdFinEquiv.symm k).2,
-   fun k => (f (x (finProdFinEquiv.symm k).1)).property (finProdFinEquiv.symm k).2⟩
-
-@[simp] lemma concatenate_slot {X : Type*} {t n : ℕ} (f : X → Cube t n)
-    (u : ℕ) (x : Fin u → X) (b : Fin u) (j : Fin n) :
-    (concatenate f u x).val (slotIndex b j) = (f (x b)).val j := by
-  simp only [concatenate, slotIndex, Equiv.symm_apply_apply]
-
-lemma boundarySlots_concatenate_sum {X : Type*} {t n : ℕ}
-    (f : X → Cube t n) (u : ℕ) (x : Fin u → X) :
-    (boundarySlots (concatenate f u x).val).card =
-      ∑ b : Fin u, (boundarySlots (f (x b)).val).card := by
-  simp only [boundarySlots, Finset.card_filter]
-  rw [← Equiv.sum_comp finProdFinEquiv, Fintype.sum_prod_type]
-  apply Finset.sum_congr rfl
-  intro b _
-  apply Finset.sum_congr rfl
-  intro j _
-  rw [show finProdFinEquiv (b, j) = slotIndex b j from rfl, concatenate_slot]
-
-lemma continuous_concatenate {X : Type*} [TopologicalSpace X] {t n : ℕ}
-    {f : X → Cube t n} (hf : Continuous f) (u : ℕ) : Continuous (concatenate f u) := by
-  apply Continuous.subtype_mk
-  apply continuous_pi
-  intro k
-  exact (continuous_apply (finProdFinEquiv.symm k).2).comp
-    (hf.subtype_val.comp (continuous_apply (finProdFinEquiv.symm k).1))
-
-def IsCompression {X : Type*} [TopologicalSpace X] {t n : ℕ}
-    (f : X → Cube t n) (a : ℝ) (u : ℕ) (G : (Fin u → X) → Cube t (u * n)) : Prop :=
-  Continuous G ∧
-  (∀ x, (a / 8) * (u : ℝ) * (n : ℝ) ≤ ((boundarySlots (G x).val).card : ℝ)) ∧
-  (∀ x b j, OnBoundary ((f (x b)).val j) →
-    (G x).val (slotIndex b j) = (f (x b)).val j)
-
-def ArbitrarilyCompresses {X : Type*} [TopologicalSpace X] {t n : ℕ}
-    (f : X → Cube t n) (a : ℝ) : Prop :=
-  ∀ U : ℕ, 1 ≤ U → ∃ u : ℕ, U ≤ u ∧
-    ∃ G : (Fin u → X) → Cube t (u * n), IsCompression f a u G
-
-theorem compression_pullback {X P : Type*} [TopologicalSpace X] [TopologicalSpace P]
-    {t n u : ℕ} {a : ℝ} {f : X → Cube t n} {fP : P → Cube t n}
-    (q : X → P) (hq : Continuous q) (hfactor : f = fP ∘ q)
-    {G : (Fin u → P) → Cube t (u * n)} (hG : IsCompression fP a u G) :
-    IsCompression f a u (G ∘ powerMap q u) := by
-  refine ⟨hG.1.comp (continuous_powerMap hq u), ?_, ?_⟩
-  · intro x
-    exact hG.2.1 (powerMap q u x)
-  · intro x b j hx
-    rw [hfactor] at hx ⊢
-    exact hG.2.2 (powerMap q u x) b j hx
-
-theorem arbitrarily_compresses_pullback {X P : Type*}
-    [TopologicalSpace X] [TopologicalSpace P]
-    {t n : ℕ} {a : ℝ} {f : X → Cube t n} {fP : P → Cube t n}
-    (q : X → P) (hq : Continuous q) (hfactor : f = fP ∘ q)
-    (hP : ArbitrarilyCompresses fP a) : ArbitrarilyCompresses f a := by
-  intro U hU
-  obtain ⟨u, hUu, G, hG⟩ := hP U hU
-  exact ⟨u, hUu, G ∘ powerMap q u, compression_pullback q hq hfactor hG⟩
-
-theorem arbitrarily_compresses_of_isEmpty {X : Type*} [TopologicalSpace X] [IsEmpty X]
-    {t n : ℕ} (f : X → Cube t n) (a : ℝ) : ArbitrarilyCompresses f a := by
-  intro U hU
-  let G : (Fin U → X) → Cube t (U * n) := fun _ => ⟨0, by simp [cubeSet]⟩
-  refine ⟨U, le_rfl, G, continuous_const, ?_, ?_⟩
-  · intro x
-    exact isEmptyElim (x ⟨0, by omega⟩)
-  · intro x
-    exact isEmptyElim (x ⟨0, by omega⟩)
-
-lemma repeated_compression {X : Type*} [TopologicalSpace X] {t n : ℕ}
-    {f F : X → Cube t n} {a : ℝ} (hF : Continuous F)
-    (hcount : ∀ x, (a / 8) * (n : ℝ) ≤ ((boundarySlots (F x).val).card : ℝ))
-    (hfix : ∀ x j, OnBoundary ((f x).val j) → (F x).val j = (f x).val j)
-    (u : ℕ) : IsCompression f a u (concatenate F u) := by
-  refine ⟨continuous_concatenate hF u, ?_, ?_⟩
-  · intro x
-    rw [boundarySlots_concatenate_sum, Nat.cast_sum]
-    calc
-      a / 8 * (u : ℝ) * (n : ℝ) = ∑ _b : Fin u, (a / 8) * (n : ℝ) := by
-        simp [Finset.sum_const, mul_comm, mul_left_comm]
-      _ ≤ ∑ b : Fin u, ((boundarySlots (F (x b)).val).card : ℝ) :=
-        Finset.sum_le_sum (fun b _ => hcount (x b))
-  · intro x b j hj
-    rw [concatenate_slot]
-    exact hfix (x b) j hj
-
-theorem arbitrarily_compresses_of_empty_locus {X : Type*} [TopologicalSpace X]
-    {t n : ℕ} {f : X → Cube t n} {L : Set (Slot t)} {a : ℝ}
-    (hf : Continuous f) (ha : 0 ≤ a) (l : Slot t) (hlL : l ∈ L) (hl : ‖l‖ < 1 / 2)
-    (hempty : torusLocus f L (sparseParameters t n (a * (n : ℝ))) = ∅) :
-    ArbitrarilyCompresses f a := by
-  let s := ⌊a * (n : ℝ)⌋₊
-  let g : X → Slots t n := fun x j => (f x).val j - l
-  have hg : Continuous g := hf.subtype_val.sub continuous_const
-  have hclose : ∀ x, ‖g x - (f x).val‖ < 1 / 2 := by
-    intro x
-    have heq : g x - (f x).val = fun _ : Fin n => -l := by
-      funext j
-      dsimp [g]
-      abel
-    rw [heq]
-    have hbound : ‖(fun _ : Fin n => -l)‖ ≤ ‖l‖ := by
-      rw [pi_norm_le_iff_of_nonneg (norm_nonneg l)]
-      intro j
-      simp
-    exact hbound.trans_lt hl
-  have havoid : ∀ x, g x ∉ sparseParameters t n (s : ℝ) := by
-    intro x hx
-    have hz : g x ∈ sparseParameters t n (a * (n : ℝ)) :=
-      ⟨hx.1, hx.2.trans (Nat.floor_le (mul_nonneg ha (Nat.cast_nonneg n)))⟩
-    have hq : (x, ⟨g x, hz⟩) ∈ torusLocus f L (sparseParameters t n (a * (n : ℝ))) := by
-      intro j
-      change (f x).val j - ((f x).val j - l) ∈ L
-      simpa only [sub_sub_cancel] using hlL
-    rw [hempty] at hq
-    exact hq
-  obtain ⟨F, hF, hcount, hfix⟩ :=
-    boundary_restoration_of_sparseParameters f g hf hg havoid hclose
-  have hreal : (a / 8) * (n : ℝ) ≤ ((s + 1 : ℕ) : ℝ) := by
-    have h0 : 0 ≤ a * (n : ℝ) := mul_nonneg ha (Nat.cast_nonneg n)
-    have hfloor := Nat.lt_floor_add_one (a * (n : ℝ))
-    dsimp [s]
-    push_cast
-    linarith
-  intro U _hU
-  refine ⟨U, le_rfl, concatenate F U, repeated_compression hF ?_ hfix U⟩
-  intro x
-  exact hreal.trans (Nat.cast_le.mpr (hcount x))
-
-lemma floor_batch_lower_bound {a : ℝ} {b : ℕ} (hab : 16 ≤ a * (b : ℝ)) :
-    3 * a * (b : ℝ) / 16 ≤ (⌊a * (b : ℝ) / 4⌋₊ : ℝ) := by
-  have hfloor := Nat.lt_floor_add_one (a * (b : ℝ) / 4)
-  linarith
-
-theorem padding_boundary_fraction {a : ℝ} {b B j : ℕ}
-    (ha : 0 ≤ a) (hab : 16 ≤ a * (b : ℝ)) (hj : 2 * B ≤ j) :
-    (a / 8) * ((b * (B + j) : ℕ) : ℝ) ≤
-      ((j * ⌊a * (b : ℝ) / 4⌋₊ : ℕ) : ℝ) := by
-  have hfloor := floor_batch_lower_bound hab
-  have hBj : 2 * (B : ℝ) ≤ (j : ℝ) := by exact_mod_cast hj
-  have hcount := mul_le_mul_of_nonneg_left hfloor (Nat.cast_nonneg j : 0 ≤ (j : ℝ))
-  have hscale := mul_le_mul_of_nonneg_left hBj
-    (mul_nonneg ha (Nat.cast_nonneg b : 0 ≤ (b : ℝ)))
-  push_cast
-  nlinarith
-
-end TorusCompression
-
-end
-
-end OAI
+import Mathlib
+
+namespace OAI
+
+/-! Boundary restoration, bundle rank bounds and elementary torus compression. -/
+
+noncomputable section
+
+namespace TorusCompression
+
+open Set Metric
+open scoped Classical
+
+abbrev Slot (t : ℕ) := Fin t → ℝ
+abbrev Slots (t n : ℕ) := Fin n → Slot t
+
+def cubeSet (t n : ℕ) : Set (Slots t n) := {x | ∀ j, ‖x j‖ ≤ 1}
+abbrev Cube (t n : ℕ) := {x : Slots t n // x ∈ cubeSet t n}
+
+def OnBoundary {t : ℕ} (v : Slot t) : Prop := ‖v‖ = 1
+
+lemma slot_unit_ball (t : ℕ) :
+    {v : Slot t | ∀ i, v i ∈ Icc (-1 : ℝ) 1} = closedBall 0 1 := by
+  ext v
+  simp only [mem_ofPred_eq, mem_Icc, mem_closedBall, dist_zero_right]
+  rw [pi_norm_le_iff_of_nonneg (by norm_num : (0 : ℝ) ≤ 1)]
+  simp only [Real.norm_eq_abs, abs_le]
+
+lemma onBoundary_iff_frontier {t : ℕ} (v : Slot t) :
+    OnBoundary v ↔ v ∈ frontier {w : Slot t | ∀ i, w i ∈ Icc (-1 : ℝ) 1} := by
+  rw [slot_unit_ball, frontier_closedBall _ (by norm_num : (1 : ℝ) ≠ 0)]
+  simp [OnBoundary]
+
+def support {t n : ℕ} (x : Slots t n) : Finset (Fin n) :=
+  Finset.univ.filter (fun j => x j ≠ 0)
+
+def Sparse {t n : ℕ} (s : ℕ) : Set (Slots t n) := {x | (support x).card ≤ s}
+
+def sparseParameters (t n : ℕ) (s : ℝ) : Set (Slots t n) :=
+  {z | (∀ j, ‖z j‖ ≤ 2) ∧ ((support z).card : ℝ) ≤ s}
+
+def boundarySlots {t n : ℕ} (x : Slots t n) : Finset (Fin n) :=
+  Finset.univ.filter (fun j => OnBoundary (x j))
+
+@[simp] lemma mem_support {t n : ℕ} (x : Slots t n) (j : Fin n) :
+    j ∈ support x ↔ x j ≠ 0 := by simp [support]
+
+@[simp] lemma mem_boundarySlots {t n : ℕ} (x : Slots t n) (j : Fin n) :
+    j ∈ boundarySlots x ↔ OnBoundary (x j) := by simp [boundarySlots]
+
+@[simp] lemma support_zero (t n : ℕ) : support (0 : Slots t n) = ∅ := by
+  ext j
+  simp
+
+lemma sparse_nonempty (t n s : ℕ) : (Sparse (t := t) (n := n) s).Nonempty := by
+  exact ⟨0, by simp [Sparse]⟩
+
+lemma sparse_eq_union (t n s : ℕ) :
+    Sparse (t := t) (n := n) s =
+      ⋃ A : {A : Finset (Fin n) // A.card ≤ s},
+        {x : Slots t n | ∀ j, j ∉ A.val → x j = 0} := by
+  ext x
+  constructor
+  · intro hx
+    refine mem_iUnion.mpr ⟨⟨support x, hx⟩, ?_⟩
+    intro j hj
+    simpa using hj
+  · intro hx
+    obtain ⟨A, hA⟩ := mem_iUnion.mp hx
+    exact (Finset.card_le_card (show support x ⊆ A.val from by
+      intro j hj
+      by_contra hjA
+      exact (mem_support x j).mp hj (hA j hjA))).trans A.property
+
+lemma isClosed_sparse (t n s : ℕ) : IsClosed (Sparse (t := t) (n := n) s) := by
+  rw [sparse_eq_union]
+  apply isClosed_iUnion_of_finite
+  intro A
+  simp only [ofPred_forall]
+  exact isClosed_iInter fun j => isClosed_iInter fun _ =>
+    isClosed_eq (continuous_apply j)
+      (continuous_const : Continuous (fun _ : Slots t n => (0 : Slot t)))
+
+lemma sparseParameters_eq_empty {t n : ℕ} {s : ℝ} (hs : s < 0) :
+    sparseParameters t n s = ∅ := by
+  apply Set.eq_empty_iff_forall_notMem.mpr
+  intro x hx
+  have hcard : (0 : ℝ) ≤ (support x).card := Nat.cast_nonneg _
+  exact (not_le_of_gt hs) (hcard.trans hx.2)
+
+lemma sparseParameters_eq_inter {t n : ℕ} {s : ℝ} (hs : 0 ≤ s) :
+    sparseParameters t n s = closedBall (0 : Slots t n) 2 ∩ Sparse ⌊s⌋₊ := by
+  ext x
+  simp only [sparseParameters, Sparse, mem_ofPred_eq, mem_inter_iff,
+    mem_closedBall, dist_zero_right]
+  rw [pi_norm_le_iff_of_nonneg (by norm_num : (0 : ℝ) ≤ 2)]
+  exact and_congr Iff.rfl (Nat.le_floor_iff hs).symm
+
+lemma isCompact_sparseParameters (t n : ℕ) (s : ℝ) :
+    IsCompact (sparseParameters t n s) := by
+  by_cases hs : 0 ≤ s
+  · rw [sparseParameters_eq_inter hs]
+    exact (isCompact_closedBall (0 : Slots t n) 2).inter_right (isClosed_sparse t n ⌊s⌋₊)
+  · rw [sparseParameters_eq_empty (lt_of_not_ge hs)]
+    exact isCompact_empty
+
+lemma sparseParameters_eq_parameterCube {t n : ℕ} {s : ℝ} (hs : (n : ℝ) ≤ s) :
+    sparseParameters t n s = {z : Slots t n | ∀ j, ‖z j‖ ≤ 2} := by
+  ext z
+  refine ⟨fun hz => hz.1, fun hz => ⟨hz, ?_⟩⟩
+  have hcard : (support z).card ≤ n := by
+    exact (Finset.card_le_card (Finset.filter_subset _ _)).trans_eq (Finset.card_fin n)
+  exact (Nat.cast_le.mpr hcard).trans hs
+
+def restorationRadius {t n : ℕ} (s : ℕ) (x : Slots t n) : ℝ :=
+  min (1 / 2) (infDist x (Sparse s) / 2)
+
+lemma continuous_restorationRadius (t n s : ℕ) :
+    Continuous (restorationRadius (t := t) (n := n) s) := by
+  unfold restorationRadius
+  exact continuous_const.min ((continuous_infDist_pt (Sparse s)).div_const 2)
+
+lemma restorationRadius_pos {t n s : ℕ} {x : Slots t n} (hx : x ∉ Sparse s) :
+    0 < restorationRadius s x := by
+  have h := (isClosed_sparse t n s).notMem_iff_infDist_pos (sparse_nonempty t n s) |>.mp hx
+  exact lt_min (by norm_num) (by positivity)
+
+lemma restorationRadius_le_half {t n s : ℕ} (x : Slots t n) :
+    restorationRadius s x ≤ 1 / 2 := min_le_left _ _
+
+lemma restorationRadius_lt_infDist {t n s : ℕ} {x : Slots t n}
+    (hx : x ∉ Sparse s) : restorationRadius s x < infDist x (Sparse s) := by
+  have h := (isClosed_sparse t n s).notMem_iff_infDist_pos (sparse_nonempty t n s) |>.mp hx
+  exact (min_le_right _ _).trans_lt (by linarith)
+
+lemma many_large_slots {t n s : ℕ} {x : Slots t n} (hx : x ∉ Sparse s) :
+    s < (Finset.univ.filter (fun j => restorationRadius s x ≤ ‖x j‖)).card := by
+  let δ := restorationRadius s x
+  have hδ : 0 < δ := restorationRadius_pos hx
+  let A : Finset (Fin n) := Finset.univ.filter (fun j => δ ≤ ‖x j‖)
+  let y : Slots t n := fun j => if j ∈ A then x j else 0
+  by_contra h
+  have hA : A.card ≤ s := Nat.le_of_not_lt h
+  have hy : y ∈ Sparse s := by
+    apply le_trans (Finset.card_le_card (show support y ⊆ A from ?_)) hA
+    intro j hj
+    by_contra hjA
+    have hnonzero := (mem_support y j).mp hj
+    exact hnonzero (by simp [y, hjA])
+  have hdist : dist x y ≤ δ := by
+    rw [dist_eq_norm, pi_norm_le_iff_of_nonneg hδ.le]
+    intro j
+    by_cases hj : j ∈ A
+    · simp [y, hj, hδ.le]
+    · have hj' : ‖x j‖ < δ := by simpa [A] using hj
+      simpa [y, hj] using hj'.le
+  have hlt := restorationRadius_lt_infDist hx
+  exact not_lt_of_ge ((infDist_le_dist_of_mem hy).trans hdist) hlt
+
+def radial {t : ℕ} (δ : ℝ) (v : Slot t) : Slot t :=
+  (max δ ‖v‖)⁻¹ • v
+
+lemma norm_radial_le_one {t : ℕ} {δ : ℝ} (hδ : 0 < δ) (v : Slot t) :
+    ‖radial δ v‖ ≤ 1 := by
+  have hden : 0 < max δ ‖v‖ := hδ.trans_le (le_max_left _ _)
+  rw [radial, norm_smul, Real.norm_eq_abs, abs_inv,
+    abs_of_pos hden, ← div_eq_inv_mul]
+  exact (div_le_one hden).mpr (le_max_right _ _)
+
+lemma radial_boundary {t : ℕ} {δ : ℝ} (hδ : 0 < δ) {v : Slot t}
+    (hv : δ ≤ ‖v‖) : OnBoundary (radial δ v) := by
+  have hn : 0 < ‖v‖ := hδ.trans_le hv
+  simp [OnBoundary, radial, max_eq_right hv, norm_smul, inv_mul_cancel₀ hn.ne']
+
+lemma radial_fixes_boundary {t : ℕ} {δ : ℝ} (hδ : δ ≤ 1) {v : Slot t}
+    (hv : OnBoundary v) : radial δ v = v := by
+  simp [radial, show ‖v‖ = 1 from hv, max_eq_right hδ]
+
+def cutoff {t : ℕ} (v : Slot t) : ℝ := max 0 (4 * ‖v‖ - 3)
+
+def blend {t : ℕ} (v w : Slot t) : Slot t :=
+  (1 - cutoff v) • w + cutoff v • v
+
+lemma cutoff_range {t : ℕ} {v : Slot t} (hv : ‖v‖ ≤ 1) :
+    0 ≤ cutoff v ∧ cutoff v ≤ 1 := by
+  refine ⟨le_max_left _ _, max_le (by norm_num) ?_⟩
+  linarith
+
+lemma blend_sub {t : ℕ} (v w : Slot t) :
+    blend v w - v = (1 - cutoff v) • (w - v) := by
+  simp only [blend, smul_sub, sub_smul, one_smul]
+  abel
+
+lemma norm_blend_sub_le {t : ℕ} {v w : Slot t} (hv : ‖v‖ ≤ 1) :
+    ‖blend v w - v‖ ≤ ‖w - v‖ := by
+  obtain ⟨h0, h1⟩ := cutoff_range hv
+  rw [blend_sub, norm_smul, Real.norm_eq_abs, abs_of_nonneg (sub_nonneg.mpr h1)]
+  exact mul_le_of_le_one_left (norm_nonneg _) (by linarith)
+
+lemma blend_nonzero {t : ℕ} {v w : Slot t} (hv : ‖v‖ ≤ 1)
+    (hvw : ‖w - v‖ < 1 / 2) (hw : w ≠ 0) : blend v w ≠ 0 := by
+  by_cases hχ : cutoff v = 0
+  · simpa [blend, hχ] using hw
+  · have hχpos : 0 < cutoff v := lt_of_le_of_ne (cutoff_range hv).1 (Ne.symm hχ)
+    have hnorm : 3 / 4 < ‖v‖ := by
+      have h := (lt_max_iff.mp hχpos).resolve_left (lt_irrefl 0)
+      linarith
+    intro hzero
+    have h := (norm_blend_sub_le hv).trans_lt hvw
+    rw [hzero, zero_sub, norm_neg] at h
+    linarith
+
+lemma blend_fixes_boundary {t : ℕ} {v w : Slot t} (hv : OnBoundary v) :
+    blend v w = v := by
+  have hχ : cutoff v = 1 := by simp [cutoff, show ‖v‖ = 1 from hv]; norm_num
+  simp [blend, hχ]
+
+lemma continuous_blend {t : ℕ} : Continuous (fun p : Slot t × Slot t => blend p.1 p.2) := by
+  unfold blend cutoff
+  fun_prop
+
+lemma continuous_blend_of {X : Type*} [TopologicalSpace X] {t : ℕ}
+    {v w : X → Slot t} (hv : Continuous v) (hw : Continuous w) :
+    Continuous (fun x => blend (v x) (w x)) := by
+  have hc : Continuous (fun x => cutoff (v x)) :=
+    continuous_const.max ((continuous_const.mul hv.norm).sub continuous_const)
+  exact ((continuous_const.sub hc).smul hw).add (hc.smul hv)
+
+theorem boundary_restoration {X : Type*} [TopologicalSpace X]
+    {t n s : ℕ} (f : X → Cube t n) (g : X → Slots t n)
+    (hf : Continuous f) (hg : Continuous g)
+    (havoid : ∀ x, g x ∉ Sparse s)
+    (hclose : ∀ x, ‖g x - (f x).val‖ < 1 / 2) :
+    ∃ G : X → Cube t n, Continuous G ∧
+      (∀ x, s + 1 ≤ (boundarySlots (G x).val).card) ∧
+      (∀ x j, OnBoundary ((f x).val j) → (G x).val j = (f x).val j) := by
+  let g' : X → Slots t n := fun x j => blend ((f x).val j) (g x j)
+  have hfj : ∀ j, Continuous (fun x => (f x).val j) := fun j =>
+    (continuous_apply j).comp hf.subtype_val
+  have hgj : ∀ j, Continuous (fun x => g x j) := fun j =>
+    (continuous_apply j).comp hg
+  have hg' : Continuous g' := by
+    apply continuous_pi
+    intro j
+    exact continuous_blend_of (hfj j) (hgj j)
+  have havoid' : ∀ x, g' x ∉ Sparse s := by
+    intro x hx
+    apply havoid x
+    apply le_trans (Finset.card_le_card (show support (g x) ⊆ support (g' x) from ?_)) hx
+    intro j hj
+    apply (mem_support _ _).mpr
+    apply blend_nonzero ((f x).property j) _ ((mem_support _ _).mp hj)
+    exact (norm_le_pi_norm (g x - (f x).val) j).trans_lt (hclose x)
+  let G : X → Cube t n := fun x =>
+    ⟨fun j => radial (restorationRadius s (g' x)) (g' x j),
+      fun j => norm_radial_le_one (restorationRadius_pos (havoid' x)) (g' x j)⟩
+  refine ⟨G, ?_, ?_, ?_⟩
+  · apply Continuous.subtype_mk
+    apply continuous_pi
+    intro j
+    have hden : ∀ x, max (restorationRadius s (g' x)) ‖g' x j‖ ≠ 0 := by
+      intro x
+      exact ne_of_gt ((restorationRadius_pos (havoid' x)).trans_le (le_max_left _ _))
+    exact (((continuous_restorationRadius t n s).comp hg').max
+      (((continuous_apply j).comp hg').norm)).inv₀ hden |>.smul
+        ((continuous_apply j).comp hg')
+  · intro x
+    apply (Nat.succ_le_of_lt (many_large_slots (havoid' x))).trans
+    apply Finset.card_le_card
+    intro j hj
+    exact (mem_boundarySlots _ _).mpr
+      (radial_boundary (restorationRadius_pos (havoid' x)) ((Finset.mem_filter.mp hj).2))
+  · intro x j hj
+    change radial (restorationRadius s (g' x)) (blend ((f x).val j) (g x j)) = _
+    rw [blend_fixes_boundary hj]
+    exact radial_fixes_boundary ((restorationRadius_le_half (g' x)).trans (by norm_num)) hj
+
+theorem boundary_restoration_of_sparseParameters {X : Type*} [TopologicalSpace X]
+    {t n s : ℕ} (f : X → Cube t n) (g : X → Slots t n)
+    (hf : Continuous f) (hg : Continuous g)
+    (havoid : ∀ x, g x ∉ sparseParameters t n (s : ℝ))
+    (hclose : ∀ x, ‖g x - (f x).val‖ < 1 / 2) :
+    ∃ G : X → Cube t n, Continuous G ∧
+      (∀ x, s + 1 ≤ (boundarySlots (G x).val).card) ∧
+      (∀ x j, OnBoundary ((f x).val j) → (G x).val j = (f x).val j) := by
+  apply boundary_restoration f g hf hg ?_ hclose
+  intro x hx
+  apply havoid x
+  refine ⟨?_, ?_⟩
+  · intro j
+    have hj := (norm_le_pi_norm (g x - (f x).val) j).trans_lt (hclose x)
+    have hfbound := (f x).property j
+    have hadd := norm_le_norm_sub_add (g x j) ((f x).val j)
+    dsimp only [Pi.sub_apply] at hj
+    linarith
+  · exact_mod_cast hx
+
+def torusLocus {Y : Type*} {t n : ℕ} (f : Y → Cube t n)
+    (L : Set (Slot t)) (Z : Set (Slots t n)) : Set (Y × Z) :=
+  {q | ∀ j, (f q.1).val j - q.2.val j ∈ L}
+
+abbrev TorusLocus {Y : Type*} {t n : ℕ} (f : Y → Cube t n)
+    (L : Set (Slot t)) (Z : Set (Slots t n)) := ↥(torusLocus f L Z)
+
+def locusDifference {Y : Type*} {t n : ℕ} {f : Y → Cube t n}
+    {L : Set (Slot t)} {Z : Set (Slots t n)} (q : TorusLocus f L Z) : Fin n → L :=
+  fun j => ⟨(f q.val.1).val j - q.val.2.val j, q.property j⟩
+
+lemma isClosed_torusLocus {Y : Type*} [TopologicalSpace Y] {t n : ℕ}
+    {f : Y → Cube t n} {L : Set (Slot t)} {Z : Set (Slots t n)}
+    (hf : Continuous f) (hL : IsClosed L) : IsClosed (torusLocus f L Z) := by
+  unfold torusLocus
+  simp only [ofPred_forall]
+  apply isClosed_iInter
+  intro j
+  apply hL.preimage
+  have hc₁ : Continuous (fun q : Y × Z => (f q.1).val j) :=
+    (continuous_apply j).comp (hf.subtype_val.comp continuous_fst)
+  have hc₂ : Continuous (fun q : Y × Z => q.2.val j) :=
+    (continuous_apply j).comp (continuous_subtype_val.comp continuous_snd)
+  exact hc₁.sub hc₂
+
+lemma isCompact_sparse_torusLocus {Y : Type*} [TopologicalSpace Y] [CompactSpace Y]
+    {t n : ℕ} {f : Y → Cube t n} {L : Set (Slot t)}
+    (hf : Continuous f) (hL : IsClosed L) (s : ℝ) :
+    IsCompact (torusLocus f L (sparseParameters t n s)) := by
+  let : CompactSpace (sparseParameters t n s) :=
+    isCompact_iff_compactSpace.mp (isCompact_sparseParameters t n s)
+  exact (isClosed_torusLocus hf hL).isCompact
+
+lemma continuous_locusDifference {Y : Type*} [TopologicalSpace Y] {t n : ℕ}
+    {f : Y → Cube t n} {L : Set (Slot t)} {Z : Set (Slots t n)}
+    (hf : Continuous f) : Continuous (locusDifference (f := f) (L := L) (Z := Z)) := by
+  apply continuous_pi
+  intro j
+  apply Continuous.subtype_mk
+  have hc₁ : Continuous (fun q : TorusLocus f L Z => (f q.val.1).val j) :=
+    (continuous_apply j).comp
+      (hf.subtype_val.comp (continuous_fst.comp continuous_subtype_val))
+  have hc₂ : Continuous (fun q : TorusLocus f L Z => q.val.2.val j) :=
+    (continuous_apply j).comp (continuous_subtype_val.comp
+      (continuous_snd.comp continuous_subtype_val))
+  exact hc₁.sub hc₂
+
+def slotBundleFiber {Y : Type*} {t n r : ℕ} {f : Y → Cube t n}
+    {L : Set (Slot t)} {Z : Set (Slots t n)}
+    (H : L → Submodule ℂ (Fin r → ℂ)) (q : TorusLocus f L Z) :
+    Submodule ℂ (Fin n → Fin r → ℂ) :=
+  Submodule.pi univ (fun j => H (locusDifference q j))
+
+@[simp] lemma mem_slotBundleFiber {Y : Type*} {t n r : ℕ} {f : Y → Cube t n}
+    {L : Set (Slot t)} {Z : Set (Slots t n)} (H : L → Submodule ℂ (Fin r → ℂ))
+    (q : TorusLocus f L Z) (v : Fin n → Fin r → ℂ) :
+    v ∈ slotBundleFiber H q ↔ ∀ j, v j ∈ H (locusDifference q j) := by
+  simp [slotBundleFiber, Submodule.mem_pi]
+
+def EmbeddedTotal {B V : Type*} [AddCommGroup V] [Module ℂ V]
+    (E : B → Submodule ℂ V) := {q : B × V // q.2 ∈ E q.1}
+
+instance {B V : Type*} [TopologicalSpace B] [TopologicalSpace V]
+    [AddCommGroup V] [Module ℂ V] (E : B → Submodule ℂ V) :
+    TopologicalSpace (EmbeddedTotal E) := inferInstanceAs (TopologicalSpace {q : B × V // q.2 ∈ E q.1})
+
+structure BundleMono {B V : Type*} [TopologicalSpace B] [TopologicalSpace V]
+    [AddCommGroup V] [Module ℂ V] (E : B → Submodule ℂ V) (K : ℕ) where
+  map : ∀ b, E b →ₗ[ℂ] (Fin K → ℂ)
+  injective : ∀ b, Function.Injective (map b)
+  continuous_total : Continuous (fun q : EmbeddedTotal E => map q.val.1 ⟨q.val.2, q.property⟩)
+
+def slotBundleFiberEquiv {Y : Type*} {t n r : ℕ} {f : Y → Cube t n}
+    {L : Set (Slot t)} {Z : Set (Slots t n)}
+    (H : L → Submodule ℂ (Fin r → ℂ)) (q : TorusLocus f L Z) :
+    slotBundleFiber H q ≃ₗ[ℂ] ((j : Fin n) → H (locusDifference q j)) where
+  toFun v j := ⟨v.val j, (mem_slotBundleFiber H q v.val).mp v.property j⟩
+  invFun v := ⟨fun j => (v j).val,
+    (mem_slotBundleFiber H q _).mpr (fun j => (v j).property)⟩
+  left_inv _ := rfl
+  right_inv _ := rfl
+  map_add' _ _ := rfl
+  map_smul' _ _ := rfl
+
+lemma finrank_slotBundleFiber {Y : Type*} {t n r p : ℕ} {f : Y → Cube t n}
+    {L : Set (Slot t)} {Z : Set (Slots t n)}
+    (H : L → Submodule ℂ (Fin r → ℂ)) (hH : ∀ l, Module.finrank ℂ (H l) = p)
+    (q : TorusLocus f L Z) : Module.finrank ℂ (slotBundleFiber H q) = p * n := by
+  rw [(slotBundleFiberEquiv H q).finrank_eq, Module.finrank_pi_fintype]
+  simp [hH, Nat.mul_comm]
+
+lemma BundleMono.finrank_le {B V : Type*} [TopologicalSpace B] [TopologicalSpace V]
+    [AddCommGroup V] [Module ℂ V] {E : B → Submodule ℂ V} {K : ℕ}
+    (M : BundleMono E K) (b : B) : Module.finrank ℂ (E b) ≤ K := by
+  have h := LinearMap.finrank_le_finrank_of_injective (M.injective b)
+  simpa using h
+
+theorem rank_le_target_of_nonempty_locus {Y : Type*} [TopologicalSpace Y]
+    {t n r p K : ℕ} {f : Y → Cube t n}
+    {L : Set (Slot t)} {Z : Set (Slots t n)}
+    (H : L → Submodule ℂ (Fin r → ℂ)) (hH : ∀ l, Module.finrank ℂ (H l) = p)
+    (M : BundleMono (slotBundleFiber H (f := f) (Z := Z)) K)
+    (hq : (torusLocus f L Z).Nonempty) : p * n ≤ K := by
+  obtain ⟨q, hq⟩ := hq
+  simpa [finrank_slotBundleFiber H hH] using M.finrank_le (⟨q, hq⟩ : TorusLocus f L Z)
+
+def slotIndex {u n : ℕ} (b : Fin u) (j : Fin n) : Fin (u * n) :=
+  finProdFinEquiv (b, j)
+
+@[simp] lemma slotIndex_val {u n : ℕ} (b : Fin u) (j : Fin n) :
+    (slotIndex b j).val = b.val * n + j.val := by
+  simp [slotIndex, finProdFinEquiv, Nat.mul_comm, Nat.add_comm]
+
+def powerMap {X Y : Type*} (q : X → Y) (u : ℕ) (x : Fin u → X) : Fin u → Y :=
+  fun b => q (x b)
+
+lemma continuous_powerMap {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
+    {q : X → Y} (hq : Continuous q) (u : ℕ) : Continuous (powerMap q u) := by
+  exact continuous_pi fun b => hq.comp (continuous_apply b)
+
+def concatenate {X : Type*} {t n : ℕ} (f : X → Cube t n) (u : ℕ)
+    (x : Fin u → X) : Cube t (u * n) :=
+  ⟨fun k => (f (x (finProdFinEquiv.symm k).1)).val (finProdFinEquiv.symm k).2,
+   fun k => (f (x (finProdFinEquiv.symm k).1)).property (finProdFinEquiv.symm k).2⟩
+
+@[simp] lemma concatenate_slot {X : Type*} {t n : ℕ} (f : X → Cube t n)
+    (u : ℕ) (x : Fin u → X) (b : Fin u) (j : Fin n) :
+    (concatenate f u x).val (slotIndex b j) = (f (x b)).val j := by
+  simp only [concatenate, slotIndex, Equiv.symm_apply_apply]
+
+lemma boundarySlots_concatenate_sum {X : Type*} {t n : ℕ}
+    (f : X → Cube t n) (u : ℕ) (x : Fin u → X) :
+    (boundarySlots (concatenate f u x).val).card =
+      ∑ b : Fin u, (boundarySlots (f (x b)).val).card := by
+  simp only [boundarySlots, Finset.card_filter]
+  rw [← Equiv.sum_comp finProdFinEquiv, Fintype.sum_prod_type]
+  apply Finset.sum_congr rfl
+  intro b _
+  apply Finset.sum_congr rfl
+  intro j _
+  rw [show finProdFinEquiv (b, j) = slotIndex b j from rfl, concatenate_slot]
+
+lemma continuous_concatenate {X : Type*} [TopologicalSpace X] {t n : ℕ}
+    {f : X → Cube t n} (hf : Continuous f) (u : ℕ) : Continuous (concatenate f u) := by
+  apply Continuous.subtype_mk
+  apply continuous_pi
+  intro k
+  exact (continuous_apply (finProdFinEquiv.symm k).2).comp
+    (hf.subtype_val.comp (continuous_apply (finProdFinEquiv.symm k).1))
+
+def IsCompression {X : Type*} [TopologicalSpace X] {t n : ℕ}
+    (f : X → Cube t n) (a : ℝ) (u : ℕ) (G : (Fin u → X) → Cube t (u * n)) : Prop :=
+  Continuous G ∧
+  (∀ x, (a / 8) * (u : ℝ) * (n : ℝ) ≤ ((boundarySlots (G x).val).card : ℝ)) ∧
+  (∀ x b j, OnBoundary ((f (x b)).val j) →
+    (G x).val (slotIndex b j) = (f (x b)).val j)
+
+def ArbitrarilyCompresses {X : Type*} [TopologicalSpace X] {t n : ℕ}
+    (f : X → Cube t n) (a : ℝ) : Prop :=
+  ∀ U : ℕ, 1 ≤ U → ∃ u : ℕ, U ≤ u ∧
+    ∃ G : (Fin u → X) → Cube t (u * n), IsCompression f a u G
+
+theorem compression_pullback {X P : Type*} [TopologicalSpace X] [TopologicalSpace P]
+    {t n u : ℕ} {a : ℝ} {f : X → Cube t n} {fP : P → Cube t n}
+    (q : X → P) (hq : Continuous q) (hfactor : f = fP ∘ q)
+    {G : (Fin u → P) → Cube t (u * n)} (hG : IsCompression fP a u G) :
+    IsCompression f a u (G ∘ powerMap q u) := by
+  refine ⟨hG.1.comp (continuous_powerMap hq u), ?_, ?_⟩
+  · intro x
+    exact hG.2.1 (powerMap q u x)
+  · intro x b j hx
+    rw [hfactor] at hx ⊢
+    exact hG.2.2 (powerMap q u x) b j hx
+
+theorem arbitrarily_compresses_pullback {X P : Type*}
+    [TopologicalSpace X] [TopologicalSpace P]
+    {t n : ℕ} {a : ℝ} {f : X → Cube t n} {fP : P → Cube t n}
+    (q : X → P) (hq : Continuous q) (hfactor : f = fP ∘ q)
+    (hP : ArbitrarilyCompresses fP a) : ArbitrarilyCompresses f a := by
+  intro U hU
+  obtain ⟨u, hUu, G, hG⟩ := hP U hU
+  exact ⟨u, hUu, G ∘ powerMap q u, compression_pullback q hq hfactor hG⟩
+
+theorem arbitrarily_compresses_of_isEmpty {X : Type*} [TopologicalSpace X] [IsEmpty X]
+    {t n : ℕ} (f : X → Cube t n) (a : ℝ) : ArbitrarilyCompresses f a := by
+  intro U hU
+  let G : (Fin U → X) → Cube t (U * n) := fun _ => ⟨0, by simp [cubeSet]⟩
+  refine ⟨U, le_rfl, G, continuous_const, ?_, ?_⟩
+  · intro x
+    exact isEmptyElim (x ⟨0, by omega⟩)
+  · intro x
+    exact isEmptyElim (x ⟨0, by omega⟩)
+
+lemma repeated_compression {X : Type*} [TopologicalSpace X] {t n : ℕ}
+    {f F : X → Cube t n} {a : ℝ} (hF : Continuous F)
+    (hcount : ∀ x, (a / 8) * (n : ℝ) ≤ ((boundarySlots (F x).val).card : ℝ))
+    (hfix : ∀ x j, OnBoundary ((f x).val j) → (F x).val j = (f x).val j)
+    (u : ℕ) : IsCompression f a u (concatenate F u) := by
+  refine ⟨continuous_concatenate hF u, ?_, ?_⟩
+  · intro x
+    rw [boundarySlots_concatenate_sum, Nat.cast_sum]
+    calc
+      a / 8 * (u : ℝ) * (n : ℝ) = ∑ _b : Fin u, (a / 8) * (n : ℝ) := by
+        simp [Finset.sum_const, mul_comm, mul_left_comm]
+      _ ≤ ∑ b : Fin u, ((boundarySlots (F (x b)).val).card : ℝ) :=
+        Finset.sum_le_sum (fun b _ => hcount (x b))
+  · intro x b j hj
+    rw [concatenate_slot]
+    exact hfix (x b) j hj
+
+theorem arbitrarily_compresses_of_empty_locus {X : Type*} [TopologicalSpace X]
+    {t n : ℕ} {f : X → Cube t n} {L : Set (Slot t)} {a : ℝ}
+    (hf : Continuous f) (ha : 0 ≤ a) (l : Slot t) (hlL : l ∈ L) (hl : ‖l‖ < 1 / 2)
+    (hempty : torusLocus f L (sparseParameters t n (a * (n : ℝ))) = ∅) :
+    ArbitrarilyCompresses f a := by
+  let s := ⌊a * (n : ℝ)⌋₊
+  let g : X → Slots t n := fun x j => (f x).val j - l
+  have hg : Continuous g := hf.subtype_val.sub continuous_const
+  have hclose : ∀ x, ‖g x - (f x).val‖ < 1 / 2 := by
+    intro x
+    have heq : g x - (f x).val = fun _ : Fin n => -l := by
+      funext j
+      dsimp [g]
+      abel
+    rw [heq]
+    have hbound : ‖(fun _ : Fin n => -l)‖ ≤ ‖l‖ := by
+      rw [pi_norm_le_iff_of_nonneg (norm_nonneg l)]
+      intro j
+      simp
+    exact hbound.trans_lt hl
+  have havoid : ∀ x, g x ∉ sparseParameters t n (s : ℝ) := by
+    intro x hx
+    have hz : g x ∈ sparseParameters t n (a * (n : ℝ)) :=
+      ⟨hx.1, hx.2.trans (Nat.floor_le (mul_nonneg ha (Nat.cast_nonneg n)))⟩
+    have hq : (x, ⟨g x, hz⟩) ∈ torusLocus f L (sparseParameters t n (a * (n : ℝ))) := by
+      intro j
+      change (f x).val j - ((f x).val j - l) ∈ L
+      simpa only [sub_sub_cancel] using hlL
+    rw [hempty] at hq
+    exact hq
+  obtain ⟨F, hF, hcount, hfix⟩ :=
+    boundary_restoration_of_sparseParameters f g hf hg havoid hclose
+  have hreal : (a / 8) * (n : ℝ) ≤ ((s + 1 : ℕ) : ℝ) := by
+    have h0 : 0 ≤ a * (n : ℝ) := mul_nonneg ha (Nat.cast_nonneg n)
+    have hfloor := Nat.lt_floor_add_one (a * (n : ℝ))
+    dsimp [s]
+    push_cast
+    linarith
+  intro U _hU
+  refine ⟨U, le_rfl, concatenate F U, repeated_compression hF ?_ hfix U⟩
+  intro x
+  exact hreal.trans (Nat.cast_le.mpr (hcount x))
+
+lemma floor_batch_lower_bound {a : ℝ} {b : ℕ} (hab : 16 ≤ a * (b : ℝ)) :
+    3 * a * (b : ℝ) / 16 ≤ (⌊a * (b : ℝ) / 4⌋₊ : ℝ) := by
+  have hfloor := Nat.lt_floor_add_one (a * (b : ℝ) / 4)
+  linarith
+
+theorem padding_boundary_fraction {a : ℝ} {b B j : ℕ}
+    (ha : 0 ≤ a) (hab : 16 ≤ a * (b : ℝ)) (hj : 2 * B ≤ j) :
+    (a / 8) * ((b * (B + j) : ℕ) : ℝ) ≤
+      ((j * ⌊a * (b : ℝ) / 4⌋₊ : ℕ) : ℝ) := by
+  have hfloor := floor_batch_lower_bound hab
+  have hBj : 2 * (B : ℝ) ≤ (j : ℝ) := by exact_mod_cast hj
+  have hcount := mul_le_mul_of_nonneg_left hfloor (Nat.cast_nonneg j : 0 ≤ (j : ℝ))
+  have hscale := mul_le_mul_of_nonneg_left hBj
+    (mul_nonneg ha (Nat.cast_nonneg b : 0 ≤ (b : ℝ)))
+  push_cast
+  nlinarith
+
+end TorusCompression
+
+end
+
+end OAI

lean/README.md

@@ -1,11 +1,11 @@
-# Lean formalizations
-
-This directory contains Lean formalizations of some results in this repository.
-They are organized in a single large library, so we recommend compiling only small portions at a time.
-
-See the [Comparator README](ComparatorChallenges/README.md) for more information about how to verify the results.
-
-## Technical note: mmap
-
-Compiling the entire library may fail if Linux's `vm.max_map_count` is too low.
-One workaround is to build Lean with the CMake option `-DMMAP=OFF`; it may also be necessary to set the environment variable `GLIBC_TUNABLES` to `glibc.malloc.mmap_max=0:glibc.malloc.arena_max=1` when running Lean or Lake.
+# Lean formalizations
+
+This directory contains Lean formalizations of some results in this repository.
+They are organized in a single large library, so we recommend compiling only small portions at a time.
+
+See the [Comparator README](ComparatorChallenges/README.md) for more information about how to verify the results.
+
+## Technical note: mmap
+
+Compiling the entire library may fail if Linux's `vm.max_map_count` is too low.
+One workaround is to build Lean with the CMake option `-DMMAP=OFF`; it may also be necessary to set the environment variable `GLIBC_TUNABLES` to `glibc.malloc.mmap_max=0:glibc.malloc.arena_max=1` when running Lean or Lake.

lean/docs/027.md

@@ -1,17 +1,17 @@
-# Potential integral density on curve character varieties
-
-The following describes the scope of the Lean formalization related to the following accompanying paper(s):
-
-- [Integral points on character varieties of curves](../../preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf)
-
-## Scope
-
-The integral-density question asks whether integral points become Zariski dense in every component of a curve character variety after one finite extension of the number field. The linked formalization proves a supporting compatibility result for its marked-surface construction. Given a proper diagram with finitely many vertices, a chosen seam with maximal adjacent facet rank and a positive-genus parent, puncture data, an existing solution over a field, and a chosen boundary occurrence, the produced marked values satisfy every seam equation.
-
-This is conditional compatibility of the produced solution. Potential integral density, including the prescribed-boundary and all-component conclusions, remains outside the selected statement.
-
-## Comparator links
-
-| Result | Comparator statement |
-| --- | --- |
-| Compatibility of the produced marked solution with every seam equation | [CharacterVarietiesAllSeamsSupport.lean](../ComparatorChallenges/CharacterVarietiesAllSeamsSupport.lean) |
+# Potential integral density on curve character varieties
+
+The following describes the scope of the Lean formalization related to the following accompanying paper(s):
+
+- [Integral points on character varieties of curves](../../preprints/Integral-points-on-character-varieties-of-curves-September-25-2026/paper.pdf)
+
+## Scope
+
+The integral-density question asks whether integral points become Zariski dense in every component of a curve character variety after one finite extension of the number field. The linked formalization proves a supporting compatibility result for its marked-surface construction. Given a proper diagram with finitely many vertices, a chosen seam with maximal adjacent facet rank and a positive-genus parent, puncture data, an existing solution over a field, and a chosen boundary occurrence, the produced marked values satisfy every seam equation.
+
+This is conditional compatibility of the produced solution. Potential integral density, including the prescribed-boundary and all-component conclusions, remains outside the selected statement.
+
+## Comparator links
+
+| Result | Comparator statement |
+| --- | --- |
+| Compatibility of the produced marked solution with every seam equation | [CharacterVarietiesAllSeamsSupport.lean](../ComparatorChallenges/CharacterVarietiesAllSeamsSupport.lean) |

lean/docs/049.md

@@ -3,9 +3,12 @@
 The following describes the scope of the Lean formalization related to the following accompanying paper(s):
 
 - [An explicit noncoordinate polynomial with affine three-space zero fibre](../../preprints/An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026/paper.pdf)
+- [A stable coordinate that is not a coordinate in four variables](../../preprints/A-stable-coordinate-that-is-not-a-coordinate-in-four-variables-October-5-2026/stable-coordinate-four-variables.pdf)
 
 ## Scope
 
+The stable coordinate conjecture predicts that a polynomial that becomes a coordinate after adjoining variables was already a coordinate. The formalization gives an explicit degree-five polynomial in four complex variables that becomes a coordinate after adjoining one variable but is not a coordinate in four variables. Every fiber is isomorphic to affine three-space, yet no fiber can be carried to a coordinate hyperplane by an ambient polynomial automorphism.
+
 The Abhyankar–Sathaye conjecture predicts that a polynomial defining an affine-space quotient must be an ambient coordinate. For every $n\ge4$, the formalized counterexample gives $F\in\mathbb C[x_1,\ldots,x_n]$ with quotient $\mathbb C[x_1,\ldots,x_n]/(F)\cong\mathbb C[y_1,\ldots,y_{n-1}]$, although $F$ is not a coordinate.
 
 A companion gives $n-1$ commuting, locally nilpotent derivations, linearly independent over the polynomial ring, whose common kernel is generated by the construction's explicit polynomial and contains no ambient coordinate. The extra $n-4$ derivations are ordinary partial derivatives in the added variables. The three-variable case is not covered.
@@ -16,3 +19,4 @@ A companion gives $n-1$ commuting, locally nilpotent derivations, linearly indep
 | --- | --- |
 | Noncoordinate polynomial in every dimension at least four | [AbhyankarSathaye.lean](../ComparatorChallenges/AbhyankarSathaye.lean) |
 | Commuting locally nilpotent derivations | [CommutingDerivations.lean](../ComparatorChallenges/CommutingDerivations.lean) |
+| Degree-five stable noncoordinate with nonrectifiable affine-space fibers | [StableCoordinateFour.lean](../ComparatorChallenges/StableCoordinateFour.lean) |