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Actual endpoint character products

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

proven

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    Natural Language Statement

    Lemma

    Reindexing the actual channels identifies endpoint group products and their binary channel value.

    Concept map
    70 concepts
    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 endpoint character productsActual 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 finiteindicesActual 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 spacesWalsh 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 sign_sum proven

    2 tensor_character_groups proven

    3 tensor_character_value proven

    Lean source view on GitHub

    1import Lax342547.TensorCharacters
    2import Lax342547.ChannelColumns
    3import Lax342547.Walsh
    4
    5/-!
    6---
    7title: Actual endpoint character products
    8type: lemma
    9---
    10Reindexing the actual channels identifies endpoint group products and their binary channel value.
    11-/
    12
    13namespace Lax342547.ChannelGroups
    14
    15open Lax342547.MomentSpace Lax342547.Walsh Lax342547.TensorCharacters
    16open Lax342547.ChannelCharacters Lax342547.ChannelColumns
    17open scoped BigOperators
    18
    19axiom sign_sum {ι : Type} [Fintype ι] (z : ι → Binary) : sign (∑ i, z i) = ∏ i,sign (z i)
    20
    21axiom tensor_character_groups {e d I J : Type} [Fintype e] [Fintype d] [Fintype I] [Fintype J]
    22 (A : e → Matrix I J Binary) (h : ℕ)
    23 (c : (e × d) × Fin h → (I → Binary) × (J → Binary)) :
    24 tensorCharacter h (fun i : e × d => A i.1) c =
    25 ∏ j : d,tensorCharacter h A (fun i => c ((i.1,j),i.2))
    26
    27noncomputable def channelValue {I J : Type} [Fintype I] [Fintype J] (h : ℕ)
    28 (A : Matrix I J Binary) (X : Matrix I (Fin h) Binary) (Y : Matrix J (Fin h) Binary) : Binary :=
    29 ∑ k,dotProduct (fun i => X i k) (A.mulVec (fun j => Y j k))
    30
    31axiom tensor_character_value {e N : Type} [Fintype e] [Fintype N] (h : ℕ)
    32 (A : e → Matrix N N Binary)
    33 (z : e → Matrix N (Fin h) Binary × Matrix N (Fin h) Binary) :
    34 tensorCharacter h A (columnsEquiv h z) = sign (∑ i,channelValue h (A i) (z i).1 (z i).2)
    35
    36end Lax342547.ChannelGroups
    37
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