lean/OAI/Probability/CriticalPercolation/Main.lean

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