Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Actual raw frame channel phase tails

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Lemma

    Reindexing the actual frame channels transfers independent channel character tails to the raw frame law with the explicit joint density factor.

    Concept map
    69 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 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 ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 6 statements. Each proof establishes one of them relative to its assumptions.

    1 columns_uniform_weights proven

    2 no_cover_raw_phase_tail proven

    3 pmf_weights_map proven

    4 push_cell_mass proven

    5 raw_channel_event_cap proven

    6 raw_channel_point_cap proven

    Lean source view on GitHub

    1import Lax342547.ChannelDensity
    2import Lax342547.NoCoverMoments
    3import Lax342547.RealCellLaws
    4
    5/-!
    6---
    7title: Actual raw frame channel phase tails
    8type: lemma
    9---
    10Reindexing the actual frame channels transfers independent channel character tails to the raw frame law with the explicit joint density factor.
    11-/
    12
    13namespace Lax342547.ChannelColumns
    14
    15open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.ChannelDensity Lax342547.ChannelCharacters
    16open Lax342547.RealCellLaws Lax342547.FiniteSampling Lax342547.TensorCharacters
    17open Lax342547.PushforwardWalsh Lax342547.RetainedImages
    18open Lax342547.RelativeEntropy Lax342547.ComponentSpaces Lax342547.ComponentDuals Lax342547.CoverProjection
    19open scoped BigOperators
    20
    21noncomputable def columnsEquiv {e N : Type} (h : ℕ) :
    22 (e → Matrix N (Fin h) Binary × Matrix N (Fin h) Binary) ≃
    23 (e × Fin h → (N → Binary) × (N → Binary)) where
    24 toFun z := fun a => (fun i => (z a.1).1 i a.2,fun i => (z a.1).2 i a.2)
    25 invFun c := fun a => (fun i j => (c (a,j)).1 i,fun i j => (c (a,j)).2 i)
    26 left_inv _ := rfl
    27 right_inv _ := rfl
    28
    29axiom columns_uniform_weights {e N : Type} [Fintype e] [Fintype N] [DecidableEq e] [DecidableEq N] (h : ℕ)
    30 (z : e → Matrix N (Fin h) Binary × Matrix N (Fin h) Binary) :
    31 weights (PMF.uniformOfFintype (e → Matrix N (Fin h) Binary × Matrix N (Fin h) Binary)) z =
    32 productLaw (fun _ : e × Fin h => channelLaw) (columnsEquiv h z)
    33
    34axiom pmf_weights_map {Ω X : Type} [Fintype Ω] (p : PMF Ω) (f : Ω → X) (x : X) :
    35 weights (p.map f) x = push (weights p) f x
    36
    37axiom push_cell_mass {Ω X : Type} [Fintype Ω] [Fintype X]
    38 (ρ : Ω → ℝ) (f : Ω → X) (P : X → Prop) :
    39 cellMass (push ρ f) P = cellMass ρ (fun o => P (f o))
    40
    41axiom raw_channel_point_cap {e B N : Type} [Fintype e] [Fintype B] [Fintype N]
    42 [DecidableEq e] [DecidableEq N] (h : ℕ) (E : Matrix B B Binary)
    43 [Nonempty (Frame B (Fin h) N E)] (hN : 2*h+1 ≤ Fintype.card N)
    44 (z : e → Matrix N (Fin h) Binary × Matrix N (Fin h) Binary) :
    45 weights ((Lax342547.RawLaw.uniformLaw E).map (fun o a => channels (o a))) z ≤
    46 (2 : ℝ)^(Fintype.card e*(h*h+2))*
    47 productLaw (fun _ : e × Fin h => channelLaw) (columnsEquiv h z)
    48
    49axiom raw_channel_event_cap {e B N : Type} [Fintype e] [Fintype B] [Fintype N]
    50 [DecidableEq e] [DecidableEq N] (h : ℕ) (E : Matrix B B Binary)
    51 [Nonempty (Frame B (Fin h) N E)] (hN : 2*h+1 ≤ Fintype.card N)
    52 (P : (e × Fin h → (N → Binary) × (N → Binary)) → Prop) :
    53 cellMass (weights (Lax342547.RawLaw.uniformLaw E))
    54 (fun o => P (columnsEquiv h (fun a => channels (o a)))) ≤
    55 (2 : ℝ)^(Fintype.card e*(h*h+2))*cellMass (productLaw (fun _ : e × Fin h => channelLaw)) P
    56
    57axiom no_cover_raw_phase_tail {e B N Ω : Type} [Fintype e] [Fintype B] [Fintype N] [Fintype Ω]
    58 [DecidableEq e] [DecidableEq N] (E : Matrix B B Binary)
    59 (μ : Ω → ℝ) (A : Ω → e → Matrix N N Binary) (ψ : Ω → ℝ) (r h t k : ℕ) (p θ : ℝ)
    60 [Nonempty (Frame B (Fin h) N E)] (hN : 2*h+1 ≤ Fintype.card N)
    61 (hμ : Probability μ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1) (hr : 1 ≤ r)
    62 (hA : ∀ x, ∑ i, (A x i).rank ≤ r) (hψ : ∀ x, |ψ x| ≤ 1)
    63 (hsize : 2*t = 2^k) (hk : 100*r+12 ≤ k) (hθ : 0 < θ)
    64 (hcover : ∀ S : Submodule Binary (e → N → Binary),
    65 ∀ T : Submodule Binary (Module.Dual Binary (e → N → Binary)),
    66 Componentwise S → DualComponentwise T →
    67 (Module.finrank Binary S : ℝ) ≤ (r : ℝ)*(2*t) →
    68 (Module.finrank Binary T : ℝ) ≤ (r : ℝ)*(2*t) →
    69 cellMass μ (fun x => projection (LinearMap.piMap (fun i => (A x i).mulVecLin)) S T = 0) ≤ p) :
    70 cellMass (weights (Lax342547.RawLaw.uniformLaw E))
    71 (fun o => θ ≤ |∑ x, μ x*ψ x*tensorCharacter h (A x) (columnsEquiv h (fun a => channels (o a)))|) ≤
    72 (2 : ℝ)^(Fintype.card e*(h*h+2))*
    73 (((4 : ℝ)^(2*t)*p^(2^(k-(100*r+10)))+1/(2 : ℝ)^(h*2^(k-(100*r+12))))/θ^(2*t))
    74
    75end Lax342547.ChannelColumns
    76
    Show ProofShow ProofShow ProofShow ProofShow ProofShow Proof
    Builds on
    Used by
    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…