import OAI.Combinatorics.EditApproximation.Geometry.ComparisonLift
import OAI.Combinatorics.EditApproximation.Refinement.StageTerminal
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalRefinementSpatial
import OAI.Combinatorics.EditApproximation.Numerics.RationalWideSpatial
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalRefinementInvariants
import OAI.Combinatorics.EditApproximation.Refinement.ComputedBandSpacing
import OAI.Combinatorics.EditApproximation.Geometry.UnskippedBands
import OAI.Combinatorics.EditApproximation.Numerics.RationalWideStep
import OAI.Combinatorics.EditApproximation.Refinement.StageComparison
import OAI.Combinatorics.EditApproximation.Geometry.OnlineActionDraw
import OAI.Combinatorics.EditApproximation.Geometry.ComparisonPhysical
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalLeaves
import OAI.Combinatorics.EditApproximation.Refinement.ComputedComparisonBand
import OAI.Combinatorics.EditApproximation.Geometry.BandRounding
import OAI.Combinatorics.EditApproximation.Geometry.BandEnvelopes
import OAI.Combinatorics.EditApproximation.Geometry.BandGuards
import OAI.Combinatorics.EditApproximation.Refinement.StageWarmup
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSpatial
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalWarmupSlack
import OAI.Combinatorics.EditApproximation.Geometry.BandParameters
import OAI.Combinatorics.EditApproximation.Geometry.ComparisonBandCoverage
import OAI.Combinatorics.EditApproximation.Geometry.RepresentativeBands
import OAI.Combinatorics.EditApproximation.Geometry.PrefixComparison
import OAI.Combinatorics.EditApproximation.Refinement.ComputedBands
import OAI.Combinatorics.EditApproximation.Geometry.InitialSeedTable
import OAI.Combinatorics.EditApproximation.Geometry.ComparisonParameters
import OAI.Combinatorics.EditApproximation.Numerics.RationalPrefix
import OAI.Combinatorics.EditApproximation.Coarse.CoarseArrayEval
import OAI.Combinatorics.EditApproximation.Coarse.CoarseGlobalSeed
import OAI.Combinatorics.EditApproximation.Geometry.WideBellman
import OAI.Combinatorics.EditApproximation.Geometry.PrefixCosts
import OAI.Combinatorics.EditApproximation.Geometry.ComparisonBounds
import OAI.Combinatorics.EditApproximation.Geometry.StateDomain
import OAI.Combinatorics.EditApproximation.Geometry.SharedActions
import OAI.Combinatorics.EditApproximation.Geometry.SourcePrefixes
import OAI.Combinatorics.EditApproximation.Geometry.ComparisonTree
import OAI.Combinatorics.EditApproximation.Geometry.RoundedComparison
import OAI.Combinatorics.EditApproximation.Refinement.ComputedSeed
import OAI.Combinatorics.EditApproximation.Coarse.CoarseParameters
import OAI.Combinatorics.EditApproximation.Coarse.CoarsePrecision
import OAI.Combinatorics.EditApproximation.Coarse.CoarseInitialWork
import OAI.Combinatorics.EditApproximation.Geometry.Accuracy
import OAI.Combinatorics.EditApproximation.Probability.Amplification
import OAI.Combinatorics.EditApproximation.Geometry.ActionLower
import OAI.Combinatorics.EditApproximation.Geometry.ActionVectors
import OAI.Combinatorics.EditApproximation.Geometry.ActualActionValues
import OAI.Combinatorics.EditApproximation.Geometry.AlignedStates
import OAI.Combinatorics.EditApproximation.Geometry.AlignmentSplitting
import OAI.Combinatorics.EditApproximation.Geometry.Bands
import OAI.Combinatorics.EditApproximation.Geometry.Bellman
import OAI.Combinatorics.EditApproximation.Geometry.BandRegret
import OAI.Combinatorics.EditApproximation.Geometry.BalancedIntervals
import OAI.Combinatorics.EditApproximation.Geometry.BalancedWidths
import OAI.Combinatorics.EditApproximation.Geometry.ChildIncrements
import OAI.Combinatorics.EditApproximation.Coarse.CoarseAccuracy
import OAI.Combinatorics.EditApproximation.Coarse.CoarseEstimator
import OAI.Combinatorics.EditApproximation.Coarse.CoarseSampling
import OAI.Combinatorics.EditApproximation.Coarse.CoarseProbability
import OAI.Combinatorics.EditApproximation.Coarse.CoarseTransforms
import OAI.Combinatorics.EditApproximation.Coarse.CoarseWork
import OAI.Combinatorics.EditApproximation.Coarse.CoarsePadding
import OAI.Combinatorics.EditApproximation.Coarse.CoarseInitial
import OAI.Combinatorics.EditApproximation.Coarse.CoarseSeed
import OAI.Combinatorics.EditApproximation.Geometry.ComparisonAccounting
import OAI.Combinatorics.EditApproximation.Probability.Concentration
import OAI.Combinatorics.EditApproximation.Geometry.Cones
import OAI.Combinatorics.EditApproximation.Geometry.ConeLocality
import OAI.Combinatorics.EditApproximation.Geometry.Connections
import OAI.Combinatorics.EditApproximation.Geometry.Curvature
import OAI.Combinatorics.EditApproximation.Geometry.DomainCounts
import OAI.Combinatorics.EditApproximation.Geometry.DomainExponential
import OAI.Combinatorics.EditApproximation.Probability.Exact
import OAI.Combinatorics.EditApproximation.Probability.ExactSampling
import OAI.Combinatorics.EditApproximation.Probability.ExpectedRegret
import OAI.Combinatorics.EditApproximation.Geometry.FirstCrossing
import OAI.Combinatorics.EditApproximation.Probability.FiniteAssignments
import OAI.Combinatorics.EditApproximation.Geometry.GroupSampling
import OAI.Combinatorics.EditApproximation.Geometry.Grid
import OAI.Combinatorics.EditApproximation.Geometry.Leaves
import OAI.Combinatorics.EditApproximation.Geometry.LocalGrid
import OAI.Combinatorics.EditApproximation.Geometry.LocalInitialCones
import OAI.Combinatorics.EditApproximation.Geometry.InitialCones
import OAI.Combinatorics.EditApproximation.Geometry.InitialConeGrid
import OAI.Combinatorics.EditApproximation.Geometry.InitialConeTable
import OAI.Combinatorics.EditApproximation.Probability.Median
import OAI.Combinatorics.EditApproximation.Numerics.Numerics
import OAI.Combinatorics.EditApproximation.Numerics.NumericalWork
import OAI.Combinatorics.EditApproximation.Geometry.NodeRegret
import OAI.Combinatorics.EditApproximation.Geometry.ParameterSlack
import OAI.Combinatorics.EditApproximation.Geometry.Parameters
import OAI.Combinatorics.EditApproximation.Geometry.PartitionIntervals
import OAI.Combinatorics.EditApproximation.Numerics.RationalRefinement
import OAI.Combinatorics.EditApproximation.Probability.SampledRegret
import OAI.Combinatorics.EditApproximation.Geometry.StateGeometry
import OAI.Combinatorics.EditApproximation.Probability.ThresholdMoments
import OAI.Combinatorics.EditApproximation.Geometry.TreeAccounting
import OAI.Combinatorics.EditApproximation.Geometry.TreeParameters
import OAI.Combinatorics.EditApproximation.Geometry.Work
import OAI.Combinatorics.EditApproximation.Geometry.WeightedTreeWork
import OAI.Combinatorics.EditApproximation.Geometry.WideActions
import OAI.Combinatorics.EditApproximation.Geometry.BandGroupSampling
import OAI.Combinatorics.EditApproximation.Geometry.CellGroupSelection
import OAI.Combinatorics.EditApproximation.Geometry.ComparisonLiftBounds
import OAI.Combinatorics.EditApproximation.Refinement.ComputedGroupAccuracy
import OAI.Combinatorics.EditApproximation.Refinement.ComputedGroupNumerical
import OAI.Combinatorics.EditApproximation.Geometry.GroupCells
import OAI.Combinatorics.EditApproximation.Geometry.GroupNumerical
import OAI.Combinatorics.EditApproximation.Geometry.GroupSmallConstants
import OAI.Combinatorics.EditApproximation.Geometry.ListBandRegret
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalGroupAccuracy
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalOnlineBand
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalNodeRegret
import OAI.Combinatorics.EditApproximation.Refinement.StageComparisonEnergy
import OAI.Combinatorics.EditApproximation.Refinement.StageParameterBounds

import OAI.Combinatorics.EditApproximation.Geometry.ComparisonLiftStep
import OAI.Combinatorics.EditApproximation.Refinement.StageComparisonCharges
import OAI.Combinatorics.EditApproximation.Refinement.StageComparisonBad
import OAI.Combinatorics.EditApproximation.Geometry.BandDimension
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalUnskippedComparison
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalTreeRegret
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalGenericRegret
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalComparisonBand
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalTreeGeneric
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalTreeExpectation
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalExpectedRegret
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalParentDraw
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalRefinementLocality
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalResampledRegret
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalConditionalNode
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSelectionProgram
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSelectionResampling
import OAI.Combinatorics.EditApproximation.Probability.FiniteResampling
import OAI.Combinatorics.EditApproximation.Refinement.StageComparisonPrediction
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalTreeNear
import OAI.Combinatorics.EditApproximation.Geometry.UnskippedRegretBudget
import OAI.Combinatorics.EditApproximation.Refinement.ComputedStageParameters
import OAI.Combinatorics.EditApproximation.Refinement.ComputedStageTime
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalBoundsAllSeeds
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalTreeBudget
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalProgramLaw
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalProgramRefreshWiring
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalProgramNodeRegret
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalProgramComparisonRegret
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalProgramAccuracy
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalOnlineLocality
import OAI.Combinatorics.EditApproximation.Geometry.RefinementErrorTerms
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalDenominator
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalValueBits
import OAI.Combinatorics.EditApproximation.Numerics.RationalPruning
import OAI.Combinatorics.EditApproximation.Geometry.RefinementConeLocality
import OAI.Combinatorics.EditApproximation.Geometry.LocalCells
import OAI.Combinatorics.EditApproximation.Geometry.LocalRefinementCones
import OAI.Combinatorics.EditApproximation.Geometry.GroupSamplingWork
import OAI.Combinatorics.EditApproximation.Geometry.LocalGroupCones
import OAI.Combinatorics.EditApproximation.Geometry.RoundedGroupStates
import OAI.Combinatorics.EditApproximation.Geometry.DyadicQueryWindow
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSubtree
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSubtreeMass
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSubtreeCharges
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSubtreeRegret
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSubtreeExpectation
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSubtreeBad
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSubtreeGeneric
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSubtreePrediction
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSubtreeNear
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSubtreeBudget
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalProgramSubtreeRegret
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalProgramStateAccuracy
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSeedMedian
import OAI.Combinatorics.EditApproximation.Geometry.ActualRoundedGroupInputs
import OAI.Combinatorics.EditApproximation.Geometry.SharedGroupInputBudget
import OAI.Combinatorics.EditApproximation.Geometry.SeedPassNumerics
import OAI.Combinatorics.EditApproximation.Geometry.SeedFactorSchedule
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalQueryBudget
import OAI.Combinatorics.EditApproximation.Geometry.LocalGroupBudget
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSeedPass
import OAI.Combinatorics.EditApproximation.Probability.AdaptiveSeedLaw
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSeedTables
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalSeedSchedule
import OAI.Combinatorics.EditApproximation.Binary.BinaryMemo
import OAI.Combinatorics.EditApproximation.Geometry.CurrentQueryInputs
import OAI.Combinatorics.EditApproximation.Geometry.GroupInputDependence
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalLocalGroupProgram
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalLocalWarmup
import OAI.Combinatorics.EditApproximation.Probability.FiniteSeedFinal
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalCoarseSeedLaw
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalFinalAccuracy
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalConditionalAccuracy
import OAI.Combinatorics.EditApproximation.Geometry.LocalWideStep
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalLocalRefinement
import OAI.Combinatorics.EditApproximation.Refinement.PhysicalLocalRefinementEquality
import OAI.Combinatorics.EditApproximation.Refinement.ComputedCurrentQueryBudget

import OAI.Combinatorics.EditApproximation.AlmostLinear

/-!
# Almost-linear approximation of edit distance

A uniform binary randomized approximation scheme attains success probability
at least two thirds, returns zero on equal strings, uses almost-linear expected
work for each fixed binary accuracy, and has polynomial space on every outcome.
Finite Bernstein concentration discharges the hypotheses of its conditional
accuracy and work bounds.
-/