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Exact probabilities for independent protected channel images

Lax342547.ProtectedChannels · concepts/Lax342547/ProtectedChannels.lean · lax-342547

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    Natural Language Statement

    Lemma

    An actual independent witness tuple has uniform transpose images, giving the exact probability that every image lies in a prescribed protected coefficient space.

    Concept map
    71 concepts; 10 descendants hidden
    100%
    Actual projected span deficitSampling after exposed coordinatesAdaptive sampling on disjoint coordinatesetsUnion over adaptive component coversExact-image bounds for independent affinecolumnsExact uniform bilinear character meanWalsh operator bounds with explicit bilinearrankRank of a lifted tensor sumIndependent binary channel charactersActual raw frame channel phase tailsJoint channel law and itsindependent-column densityActual channel moments and phase tailsCombined column and row mode exposureRows of diagonal tensor mapsMoment estimates for actual componenttensor phasesComponentwise mode spaces and diagonaltensorsExact conditioning costs and recovery offinite probability massesEntropy progress for residual pair lawsIndependence of distinct sample positionsJoin-stable classes of mode coversCount tensors killed by exposureActual exposure cover projectionRank of a diagonal familyDyadic span deficit estimatesA small dyadic tail scaleActual dyadic deficit recurrenceEntropy along feasible mixture lines,including new supportFull feasible support and finite informationprojectionProjection removes the exposed termsCount tested index occurrencesFinite linear images and their uniform-lawdensity boundsFinite even moment expansionFinite independent sampling and vertexexception tailsAmbient symmetries and frame marginalsGram-conditioned columns and theirrank-failure probabilityTwo-sided Gram normalization forindividually injective framesGreedy mode space exposureSmall actual greedy exposure tailsNo-cover rank growth on arbitrary finiteindicesUniform injective frames and channeltranspose failureActual deficits indexed by a distinct listFinite list tail statisticsDimension deficits after an arbitrary modemapExposed mode dimension budgetBoolean point moments with restricted basecoordinatesFinite moment probability and rank splitA low rank sum supplies an actual adaptivecoverNo-cover phase momentsRank growth without an adaptive modecoverSquared restriction cost for independent uniteventsFinite exposure partitionsProjected nonzero terms in the actualremaining sumDimensions of projected mode spacesExact probabilities for independent protectedchannel imagesWalsh bounds for independent image lawsand separated phasesRank of a tensor killed in two quotientspacesRank loss under two restrictionsRaw matrix frames and their tensorrealizationThe finite uniform raw-vertex lawOriginal retained-cell laws from finite PMFsFinite relative entropy and support costsRank loss under restriction of a bilinear formImage caps inside original retained cellsOne exposure controls both modesMonotonicity of span deficitsMode space span deficitsNonzero tensor count from span deficitsRank of an actual linear map sumCharacters of all independent tensor channelsAdmissible pair lawsOrthogonality and finite Walsh correlationbounds
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 independent_protected_probability proven

    2 protected_columns_card proven

    3 protected_probability_rank_bound proven

    Lean source view on GitHub

    1import Lax342547.InjectiveFrames
    2import Lax342547.ChannelColumns
    3import Lax342547.FiniteLinearLaw
    4
    5/-!
    6---
    7title: Exact probabilities for independent protected channel images
    8type: lemma
    9---
    10An actual independent witness tuple has uniform transpose images, giving the exact probability that every image lies in a prescribed protected coefficient space.
    11-/
    12
    13namespace Lax342547.ProtectedChannels
    14
    15open Lax342547.MomentSpace Lax342547.RetainedImages Lax342547.RealCellLaws
    16open scoped BigOperators ENNReal
    17
    18noncomputable def columnSpaceEquiv {I H : Type} [Fintype I] [Fintype H]
    19 (S : Submodule Binary (H → Binary)) :
    20 {Z : Matrix H I Binary // ∀ i,(fun h => Z h i) ∈ S} ≃ (I → S) where
    21 toFun Z := fun i => ⟨fun h => Z.val h i,Z.property i⟩
    22 invFun v := ⟨fun h i => (v i).val h,fun i => (v i).property⟩
    23 left_inv _ := rfl
    24 right_inv _ := rfl
    25
    26axiom protected_columns_card {I H : Type} [Fintype I] [Fintype H]
    27 (S : Submodule Binary (H → Binary)) :
    28 Nat.card {Z : Matrix H I Binary // ∀ i,(fun h => Z h i) ∈ S} =
    29 2^(Module.finrank Binary S*Fintype.card I)
    30
    31axiom independent_protected_probability {I H N : Type}
    32 [Fintype I] [Fintype H] [Fintype N] [DecidableEq I] [DecidableEq H] [DecidableEq N]
    33 (D : Matrix N I Binary) (hD : Function.Injective D.mulVec)
    34 (S : Submodule Binary (H → Binary)) :
    35 cellMass (weights (PMF.uniformOfFintype (Matrix N H Binary)))
    36 (fun Y => ∀ i,(fun h => (Y.transpose*D) h i) ∈ S) =
    37 (2 : ℝ)^(Module.finrank Binary S*Fintype.card I)/(2 : ℝ)^(Fintype.card H*Fintype.card I)
    38
    39axiom protected_probability_rank_bound {I H N : Type}
    40 [Fintype I] [Fintype H] [Fintype N] [DecidableEq I] [DecidableEq H] [DecidableEq N]
    41 (D : Matrix N I Binary) (hD : Function.Injective D.mulVec)
    42 (S : Submodule Binary (H → Binary)) (K : ℕ) (hS : Module.finrank Binary S ≤ K) :
    43 cellMass (weights (PMF.uniformOfFintype (Matrix N H Binary)))
    44 (fun Y => ∀ i,(fun h => (Y.transpose*D) h i) ∈ S) ≤
    45 (2 : ℝ)^(K*Fintype.card I)/(2 : ℝ)^(Fintype.card H*Fintype.card I)
    46
    47end Lax342547.ProtectedChannels
    48
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