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Reciprocal raw-frame orientation and dual pin avoidance

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

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    Natural Language Statement

    Lemma

    Swapping primal and dual frame data is an exact transpose equivalence, preserves uniform density caps, and supplies the reciprocal conditioned primal-span avoidance bound.

    Concept map
    79 concepts; 3 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 marginalsReciprocal raw-frame orientation and dualpin avoidanceGram-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-matrix densityUniform injective frames and channeltranspose failureActual deficits indexed by a distinct listFinite list tail statisticsFactorization and counting of low-rankbinary matricesDimension 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 partitionsActual phase averages from small exceptionaltailsUniform primal-span avoidanceProjected 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 lawActual raw primal-span avoidanceOriginal 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 channelsCounting component tensors with boundedtotal rankUniform raw channel phase tails overbounded-rank targetsAdmissible pair lawsWhole-unit conditioned pin avoidanceOrthogonality 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 capped_transpose_frame proven

    2 conditioned_dual_span_avoidance proven

    3 uniform_equiv_push proven

    Lean source view on GitHub

    1import Lax342547.UnitSpanAvoidance
    2import Lax342547.FiniteLinearLaw
    3
    4/-!
    5---
    6title: Reciprocal raw-frame orientation and dual pin avoidance
    7type: lemma
    8---
    9Swapping primal and dual frame data is an exact transpose equivalence, preserves uniform density caps, and supplies the reciprocal conditioned primal-span avoidance bound.
    10-/
    11
    12namespace Lax342547.FrameTranspose
    13
    14open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.RealCellLaws
    15open Lax342547.PushforwardWalsh Lax342547.RetainedImages
    16open scoped BigOperators
    17
    18def transposeFrame {P H N : Type} [Fintype P] [Fintype H] [Fintype N]
    19 {E : Matrix P P Binary} (f : Frame P H N E) : Frame P H N E.transpose where
    20 P := f.Q
    21 Q := f.P
    22 X := f.Y
    23 Y := f.X
    24 plus_injective := f.minus_injective
    25 minus_injective := f.plus_injective
    26 gram := by simpa only [Matrix.transpose_mul,Matrix.transpose_transpose] using congrArg Matrix.transpose f.gram
    27 plus_annihilator := by simpa only [Matrix.transpose_mul,Matrix.transpose_transpose,Matrix.transpose_zero] using
    28 (congrArg Matrix.transpose f.minus_annihilator)
    29 minus_annihilator := by simpa only [Matrix.transpose_mul,Matrix.transpose_transpose,Matrix.transpose_zero] using
    30 (congrArg Matrix.transpose f.plus_annihilator)
    31
    32def transposeEquiv {P H N : Type} [Fintype P] [Fintype H] [Fintype N] (E : Matrix P P Binary) :
    33 Frame P H N E ≃ Frame P H N E.transpose where
    34 toFun := transposeFrame
    35 invFun := transposeFrame
    36 left_inv f := by cases f; rfl
    37 right_inv f := by cases f; rfl
    38
    39axiom uniform_equiv_push {A B : Type} [Fintype A] [Fintype B] [Nonempty A] [Nonempty B]
    40 (e : A ≃ B) (b : B) :
    41 push (weights (PMF.uniformOfFintype A)) e b = weights (PMF.uniformOfFintype B) b
    42
    43axiom capped_transpose_frame {A P H N : Type} [Fintype A] [Fintype P] [Fintype H] [Fintype N]
    44 (E : Matrix P P Binary) [Nonempty (Frame P H N E)]
    45 (α : A → ℝ) (image : A → Frame P H N E) (M : ℝ)
    46 (hcap : ∀ f,push α image f ≤ M*weights (PMF.uniformOfFintype (Frame P H N E)) f) :
    47 letI : Nonempty (Frame P H N E.transpose) := ⟨transposeFrame (Classical.choice inferInstance)⟩
    48 ∀ f,push α (fun a => transposeFrame (image a)) f ≤
    49 M*weights (PMF.uniformOfFintype (Frame P H N E.transpose)) f
    50
    51axiom conditioned_dual_span_avoidance {A P H N : Type}
    52 [Fintype A] [Fintype P] [Fintype H] [Fintype N] [DecidableEq P] [DecidableEq H] [DecidableEq N]
    53 (E : Matrix P P Binary) [Nonempty (Frame P H N E)]
    54 (α : A → ℝ) (image : A → Frame P H N E) (V : A → Submodule Binary (N → Binary))
    55 (W : Submodule Binary (N → Binary)) (C : A → Prop) (M τ : ℝ)
    56 (hα : ∀ a,0 ≤ α a) (hM : 0 ≤ M) (hτ : 0 < τ) (hC : τ ≤ cellMass α C)
    57 (hN : Fintype.card P+Fintype.card H+1 ≤ Fintype.card N)
    58 (hV : ∀ a,V a ≤ LinearMap.range (image a).Q.mulVecLin)
    59 (hcap : ∀ f,push α image f ≤ M*weights (PMF.uniformOfFintype (Frame P H N E)) f) :
    60 cellMass (Lax342547.RetainedImages.conditionalLaw α C) (fun a => ¬ Disjoint (V a) W) ≤
    61 (M/τ)*(2*((2 : ℝ)^(Fintype.card P+Fintype.card H+Module.finrank Binary W)/(2 : ℝ)^Fintype.card N))
    62
    63end Lax342547.FrameTranspose
    64
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