| import Mathlib |
| import OAI.Probability.CriticalPercolation.GraphGeometry.ObjectKernel |
| import OAI.Probability.CriticalPercolation.Clusters.ClusterObjectAvoidanceCross |
| import OAI.Probability.CriticalPercolation.Clusters.ThreeClusterAvoidanceSecond |
| import OAI.Probability.CriticalPercolation.GraphGeometry.ResidualConvertedVariance |
| import OAI.Probability.CriticalPercolation.Clusters.ThreeClusterAvoidanceGeneral |
| import OAI.Probability.CriticalPercolation.GraphGeometry.ProbIsolateThree |
| import OAI.Probability.CriticalPercolation.GraphGeometry.PhaseStepCharge |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EraseBond |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OpenGraphInsert |
| import OAI.Probability.CriticalPercolation.GraphGeometry.CutEdges |
| import OAI.Probability.CriticalPercolation.GraphGeometry.CutEdgesSubsetIncident |
| import OAI.Probability.CriticalPercolation.Gluing.ContactPivot |
| import OAI.Probability.CriticalPercolation.Exploration.ReachableIncidenceImp |
| import OAI.Probability.CriticalPercolation.Gluing.ContactExitPivot |
| import OAI.Probability.CriticalPercolation.Exploration.TranscriptWeight |
| import OAI.Probability.CriticalPercolation.GraphGeometry.UpdateKernel |
| import OAI.Probability.CriticalPercolation.GraphGeometry.ExpectedResidualAlt |
| import OAI.Probability.CriticalPercolation.GraphGeometry.ColumnHitOthersOffdiag |
| import OAI.Probability.CriticalPercolation.GraphGeometry.LastInverseDiagonal |
| import OAI.Probability.CriticalPercolation.GraphGeometry.HittingSubAvoidance |
| import OAI.Probability.CriticalPercolation.Gluing.FiniteJointGluingStrict |
| import OAI.Probability.CriticalPercolation.Gluing.FiniteJointGluing |
| import OAI.Probability.CriticalPercolation.GraphGeometry.RestrictVertices |
| import OAI.Probability.CriticalPercolation.Modular.FiniteClusterTransport |
| import OAI.Probability.CriticalPercolation.Criticality.FiniteSeedTree |
| import OAI.Probability.CriticalPercolation.Gluing.CollarDistance |
| import OAI.Probability.CriticalPercolation.Gluing.InwardTrial |
| import OAI.Probability.CriticalPercolation.Gluing.LevelContactGeometry |
| import OAI.Probability.CriticalPercolation.Gluing.StoredPrediction |
| import OAI.Probability.CriticalPercolation.Criticality.ActualGrowth |
| import OAI.Probability.CriticalPercolation.GraphGeometry.MarkConnectivity |
| import OAI.Probability.CriticalPercolation.Criticality.GrowthEstimate |
| import OAI.Probability.CriticalPercolation.Gluing.SourceRelaySeparation |
| import OAI.Probability.CriticalPercolation.Gluing.FewContacts |
| import OAI.Probability.CriticalPercolation.Gluing.ContactSelection |
| import OAI.Probability.CriticalPercolation.Gluing.FreshContactTransfer |
| import OAI.Probability.CriticalPercolation.Gluing.DistanceCollarTransfer |
| import OAI.Probability.CriticalPercolation.Gluing.SourceLift |
| import OAI.Probability.CriticalPercolation.Gluing.SourceCollarTransfer |
| import OAI.Probability.CriticalPercolation.Gluing.FreshCollarExact |
| import OAI.Probability.CriticalPercolation.Gluing.FreshCorridor |
| import OAI.Probability.CriticalPercolation.Gluing.SourceConnectivity |
| import OAI.Probability.CriticalPercolation.ProductMeasure.SourceEventLaw |
| import OAI.Probability.CriticalPercolation.Gluing.ProspectiveCorridor |
| import OAI.Probability.CriticalPercolation.Gluing.RootCorridor |
| import OAI.Probability.CriticalPercolation.ProductMeasure.SourceMissLaw |
| import OAI.Probability.CriticalPercolation.Gluing.CoordinateProspective |
| import OAI.Probability.CriticalPercolation.Gluing.ApproximateMoveEndgame |
| import OAI.Probability.CriticalPercolation.Gluing.MoveWitness |
| import OAI.Probability.CriticalPercolation.Modular.MoveTransport |
| import OAI.Probability.CriticalPercolation.Exploration.IncidenceDegree |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteAdjacent |
| import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteAdjacentMeasure |
| import OAI.Probability.CriticalPercolation.GraphGeometry.InfiniteAdjacent |
| import OAI.Probability.CriticalPercolation.Clusters.PercolationTail |
| import OAI.Probability.CriticalPercolation.Clusters.InfiniteUniqueness |
| import OAI.Probability.CriticalPercolation.Criticality.DirectionalTail |
| import OAI.Probability.CriticalPercolation.Gluing.StripMoves |
| import OAI.Probability.CriticalPercolation.GraphGeometry.StripContinuity |
| import OAI.Probability.CriticalPercolation.GraphGeometry.StripUniform |
| import OAI.Probability.CriticalPercolation.Criticality.FiniteSusceptibility |
| import OAI.Probability.CriticalPercolation.ProductMeasure.SusceptibilityMeasure |
| import OAI.Probability.CriticalPercolation.Criticality.PositiveCritical |
| import OAI.Probability.CriticalPercolation.GraphGeometry.StripCritical |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseSeed |
| import OAI.Probability.CriticalPercolation.GraphGeometry.StripCompactness |
| import OAI.Probability.CriticalPercolation.ProductMeasure.EllipseProbability |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseOrigin |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseUniform |
| import OAI.Probability.CriticalPercolation.GraphGeometry.CriticalCoordinates |
| import OAI.Probability.CriticalPercolation.Criticality.OneDimensionalCritical |
| import OAI.Probability.CriticalPercolation.GraphGeometry.GraphAction |
| import OAI.Probability.CriticalPercolation.GraphGeometry.ActionCoordinates |
| import OAI.Probability.CriticalPercolation.GraphGeometry.InducedCritical |
| import OAI.Probability.CriticalPercolation.Criticality.ComponentCritical |
| import OAI.Probability.CriticalPercolation.Nilpotent.AbelianDescent |
| import OAI.Probability.CriticalPercolation.Nilpotent.FiniteCyclicInheritance |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitQuotientTransfer |
| import OAI.Probability.CriticalPercolation.Nilpotent.StructuredVirtuallyNilpotent |
| import OAI.Probability.CriticalPercolation.Criticality.RestrictedInterpolation |
| import OAI.Probability.CriticalPercolation.Criticality.InfiniteInterpolation |
| import OAI.Probability.CriticalPercolation.Criticality.MinimalPair |
| import OAI.Probability.CriticalPercolation.Criticality.MinimalInterpolation |
| import OAI.Probability.CriticalPercolation.GraphGeometry.AdjacentLog |
| import OAI.Probability.CriticalPercolation.Criticality.InfiniteNearbyPair |
| import OAI.Probability.CriticalPercolation.Growth.SuperpolynomialContradiction |
| import OAI.Probability.CriticalPercolation.Growth.SuperpolynomialMain |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitalPower |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitalAction |
| import OAI.Probability.CriticalPercolation.Exploration.FrontierDivergence |
| import OAI.Probability.CriticalPercolation.Exploration.OpenFrontierDivergence |
| import OAI.Probability.CriticalPercolation.Gluing.UniformFreshBoundary |
| import OAI.Probability.CriticalPercolation.Modular.ModularGrowth |
| import OAI.Probability.CriticalPercolation.Criticality.ParameterLeftBounds |
| import OAI.Probability.CriticalPercolation.GraphGeometry.BallCutGeometry |
| import OAI.Probability.CriticalPercolation.Criticality.SparseLinearCritical |
| import OAI.Probability.CriticalPercolation.Criticality.MainResidualBoundary |
| import OAI.Probability.CriticalPercolation.Modular.ModularIncidence |
| import OAI.Probability.CriticalPercolation.Modular.WeightedQueryBand |
| import OAI.Probability.CriticalPercolation.Modular.TiltedMassTransport |
| import OAI.Probability.CriticalPercolation.Modular.PeakClusters |
| import OAI.Probability.CriticalPercolation.Modular.WeightedExplorationCost |
| import OAI.Probability.CriticalPercolation.Modular.WeightedDiscrepancyLimit |
| import OAI.Probability.CriticalPercolation.Modular.ModularSuperlevels |
| import OAI.Probability.CriticalPercolation.Modular.HeavyPeaks |
| import OAI.Probability.CriticalPercolation.Modular.WeightedPeakBand |
| import OAI.Probability.CriticalPercolation.Modular.GeometricPeakBand |
| import OAI.Probability.CriticalPercolation.Modular.HeavyThinning |
| import OAI.Probability.CriticalPercolation.Modular.HeavyDiscrepancy |
| import OAI.Probability.CriticalPercolation.Modular.SignedIncidenceTransport |
| import OAI.Probability.CriticalPercolation.Modular.ExpectedIncidenceTransport |
| import OAI.Probability.CriticalPercolation.Criticality.SignedCountIdentity |
| import OAI.Probability.CriticalPercolation.Criticality.AutDisplacement |
| import OAI.Probability.CriticalPercolation.Clusters.GraphErgodicity |
| import OAI.Probability.CriticalPercolation.Modular.HeavyClusterNumber |
| import OAI.Probability.CriticalPercolation.Modular.CriticalHeavyIdentity |
| import OAI.Probability.CriticalPercolation.Clusters.ForcedDisconnectContinuity |
| import OAI.Probability.CriticalPercolation.Modular.CriticalHeavyDrop |
| import OAI.Probability.CriticalPercolation.Modular.HeavyFiniteGeometry |
| import OAI.Probability.CriticalPercolation.Modular.CriticalHeavyUniqueness |
| import OAI.Probability.CriticalPercolation.Modular.MainLightBoundary |
| import OAI.Probability.CriticalPercolation.Criticality.InvariantFiniteSets |
| import OAI.Probability.CriticalPercolation.Modular.MainInfiniteLightBoundary |
| import OAI.Probability.CriticalPercolation.Modular.HeavyHarris |
| import OAI.Probability.CriticalPercolation.Modular.CriticalHeavyNull |
| import OAI.Probability.CriticalPercolation.Modular.LightTwoPointDecay |
| import OAI.Probability.CriticalPercolation.Clusters.CriticalClusterMultiplicity |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EncounterGeometry |
| import OAI.Probability.CriticalPercolation.Modular.MainAllLightBoundary |
| import OAI.Probability.CriticalPercolation.Modular.LightCeiling |
| import OAI.Probability.CriticalPercolation.Criticality.UniformDescendingFan |
| import OAI.Probability.CriticalPercolation.Criticality.LocalDescendingBlock |
| import OAI.Probability.CriticalPercolation.Criticality.IntrinsicCriticalFragility |
| import OAI.Probability.CriticalPercolation.Criticality.DomainPercolation |
| import OAI.Probability.CriticalPercolation.Modular.LightSliceFrontier |
| import OAI.Probability.CriticalPercolation.Gluing.FreshExitContinuity |
| import OAI.Probability.CriticalPercolation.Gluing.DomainFreshExit |
| import OAI.Probability.CriticalPercolation.GraphGeometry.NestedLayerGeometry |
| import OAI.Probability.CriticalPercolation.Criticality.BlockLocalization |
| import OAI.Probability.CriticalPercolation.Criticality.UniformLocalContinuity |
| import OAI.Probability.CriticalPercolation.Criticality.LayerLocality |
| import OAI.Probability.CriticalPercolation.Exploration.LayerExposure |
| import OAI.Probability.CriticalPercolation.Criticality.LayerCounting |
| import OAI.Probability.CriticalPercolation.ProductMeasure.LayerExpectation |
| import OAI.Probability.CriticalPercolation.Criticality.LayerContradiction |
| import OAI.Probability.CriticalPercolation.Criticality.MainSparseBoundary |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitalPacking |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitalFiniteGeneration |
| import OAI.Probability.CriticalPercolation.GraphGeometry.RestrictedOrbitalPacking |
| import OAI.Probability.CriticalPercolation.Modular.ModularLevelGeometry |
| import OAI.Probability.CriticalPercolation.Classification.ConcreteStructureAssembly |
| import OAI.Probability.CriticalPercolation.Classification.FaithfulOrbitalModel |
| import OAI.Probability.CriticalPercolation.GraphGeometry.OrbitalStructureTransfer |
| import OAI.Probability.CriticalPercolation.GraphGeometry.FullOrbitalStructureTransfer |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousAverage |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousGrowth |
| import OAI.Probability.CriticalPercolation.Harmonic.PolynomialHarmonicSpace |
| import OAI.Probability.CriticalPercolation.Harmonic.CoverPoincare |
| import OAI.Probability.CriticalPercolation.Harmonic.HarmonicDimensionBound |
| import OAI.Probability.CriticalPercolation.Harmonic.LazyWalkProbability |
| import OAI.Probability.CriticalPercolation.Harmonic.WalkEnergyLower |
| import OAI.Probability.CriticalPercolation.Harmonic.WalkAlmostHarmonic |
| import OAI.Probability.CriticalPercolation.Harmonic.ProjectedWalkFunctions |
| import OAI.Probability.CriticalPercolation.Harmonic.AlmostHarmonicCompactness |
| import OAI.Probability.CriticalPercolation.Criticality.VirtualIndicability |
| import OAI.Probability.CriticalPercolation.LinearGroups.SolvableSparseClassification |
| import OAI.Probability.CriticalPercolation.Harmonic.HomogeneousCaccioppoli |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousPolynomialSpace |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousBallPacking |
| import OAI.Probability.CriticalPercolation.Harmonic.HomogeneousGramScales |
| import OAI.Probability.CriticalPercolation.Harmonic.HomogeneousPoincare |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousCover |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousDimension |
| import OAI.Probability.CriticalPercolation.Harmonic.GraphWalkEnergy |
| import OAI.Probability.CriticalPercolation.Harmonic.GraphProjectedFunctions |
| import OAI.Probability.CriticalPercolation.Harmonic.GraphHarmonicCompactness |
| import OAI.Probability.CriticalPercolation.Classification.GraphCompactNormalClassification |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Set MeasureTheory ProbabilityTheory |
| open scoped Classical |
|
|
| namespace CriticalPercolation.BondGraph |
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| theorem no_percolation_at_criticality [Infinite V] |
| (hconnected : G.fullGraph.Connected) |
| (hlocal : G.LocallyFinite) |
| (hquasi : G.QuasiTransitive) |
| (hpc : G.criticalProbability < 1) : |
| G.law G.criticalProbability G.percolates = 0 := by |
| by_contra hper |
| obtain ⟨o⟩ := hconnected.nonempty |
| obtain ⟨d,hd,C,hconn,hinf,hloc,htrans,hscales,happrox,hnot⟩ := |
| G.main_failure_topological_classification_counterexample |
| hconnected hlocal hquasi hpc hper o |
| let u : MulAction.orbit G.Aut o := ⟨o,MulAction.mem_orbit_self o⟩ |
| apply hnot |
| exact GraphTopology.compact_open_virtually_nilpotent_quotient_of_sparse_polynomial |
| (G.orbitPower o (2*C+1)) hconn hloc htrans u (d+1) (fun R => by |
| obtain ⟨n,hn,hp⟩ := hscales R |
| exact ⟨n,hn,hp u⟩) |
|
|
| end CriticalPercolation.BondGraph |
|
|
| end |
| end |
|
|
| end OAI |
|
|
