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Raw accepting-family injection with exponential loss

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

proven

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    Natural Language Statement

    Lemma

    Actual raw primal frames and channel images discharge the finite sampling hypotheses; an early channel margin gives an exponential absolute accepting-pair loss.

    Concept map
    93 concepts; 3 descendants hidden
    100%
    Finite accepting-family lossesActual finite accepting-pair injectionActual projected span deficitChannel bounds with adaptive protectedcoefficient spacesSampling after exposed coordinatesAdaptive sampling on disjoint coordinatesetsUnion over adaptive component coversExact-image bounds for independent affinecolumnsFinite amplified-test averagingExact 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 pin-injection exceptionFinite 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 independence of subspacesGreedy mode space exposureSmall actual greedy exposure tailsGreedy independent failure samplesAmplified channel failures on independentpinned spacesNo-cover rank growth on arbitrary finiteindicesEarly channel choice and injection exponentbudgetsUniform 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 budgetExplicit no-cover moment parameter marginsBoolean 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 spacesExact probabilities for independent protectedchannel imagesWalsh bounds for independent image lawsand separated phasesRank of a tensor killed in two quotientspacesRank loss under two restrictionsRaw accepting-family injection withexponential lossRaw 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 cellsFinite sequential testsOne exposure controls both modesCounting all bounded-dimensional protectedspacesMonotonicity 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 avoidanceUniversal protected pin witnessesOrthogonality 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_frame_channel proven

    2 exponential_accepting_injection proven

    3 raw_minus_accepting_injection proven

    Lean source view on GitHub

    1import Lax342547.AcceptingInjection
    2import Lax342547.InjectionBudgets
    3import Lax342547.InjectiveDensity
    4
    5/-!
    6---
    7title: Raw accepting-family injection with exponential loss
    8type: lemma
    9---
    10Actual raw primal frames and channel images discharge the finite sampling hypotheses; an early channel margin gives an exponential absolute accepting-pair loss.
    11-/
    12
    13namespace Lax342547.RawAcceptingInjection
    14
    15open Lax342547.MomentSpace Lax342547.RelativeEntropy Lax342547.RetainedImages
    16open Lax342547.RealCellLaws Lax342547.PushforwardWalsh Lax342547.FiniteInjection
    17open Lax342547.RawFrames Lax342547.FrameSymmetry
    18open scoped BigOperators
    19
    20axiom capped_frame_channel {B P N H : Type} [Fintype B] [Fintype P] [Fintype N] [Fintype H]
    21 [DecidableEq N] [DecidableEq H] (E : Matrix P P Binary) [Nonempty (Frame P H N E)]
    22 (β : B → ℝ) (image : B → Frame P H N E) (channel : Frame P H N E → Matrix N H Binary)
    23 (M : ℝ) (hM : 0 ≤ M)
    24 (hcap : ∀ f,push β image f ≤ M*weights (PMF.uniformOfFintype (Frame P H N E)) f)
    25 (hraw : ∀ y,push (weights (PMF.uniformOfFintype (Frame P H N E))) channel y ≤
    26 2*weights (PMF.uniformOfFintype (Matrix N H Binary)) y) :
    27 ∀ y,push β (fun b => channel (image b)) y ≤
    28 (2*M)*weights (PMF.uniformOfFintype (Matrix N H Binary)) y
    29
    30axiom raw_minus_accepting_injection {A B I P N H : Type}
    31 [Fintype A] [Fintype B] [Fintype I] [Fintype P] [Fintype N] [Fintype H]
    32 [DecidableEq P] [DecidableEq N] [DecidableEq H]
    33 (E : Matrix P P Binary) [Nonempty (Frame P H N E)]
    34 (α : A → ℝ) (β : B → ℝ) (C : I → A → Prop) (key : B → I)
    35 (imageA : A → Frame P H N E) (imageB : B → Frame P H N E)
    36 (V : A → Submodule Binary (N → Binary)) (S : B → Submodule Binary (H → Binary))
    37 (K ℓ : ℕ) (M MB η τ : ℝ)
    38 (hα : Probability α) (hβ : Probability β) (hM : 0 ≤ M) (hMB : 0 ≤ MB)
    39 (hη : 0 ≤ η) (hτ : 0 < τ)
    40 (hN : Fintype.card P+Fintype.card H+1 ≤ Fintype.card N)
    41 (hV : ∀ a,V a ≤ LinearMap.range (imageA a).P.mulVecLin)
    42 (hdV : ∀ a,Module.finrank Binary (V a) ≤ K)
    43 (hS : ∀ b,Module.finrank Binary (S b) ≤ K)
    44 (hAcap : ∀ f,push α imageA f ≤ M*weights (PMF.uniformOfFintype (Frame P H N E)) f)
    45 (hBcap : ∀ f,push β imageB f ≤ MB*weights (PMF.uniformOfFintype (Frame P H N E)) f)
    46 (hgap : (M/τ)*(2*((2 : ℝ)^(Fintype.card P+Fintype.card H+K*ℓ)/(2 : ℝ)^Fintype.card N)) < η) :
    47 (∑ b,β b*cellMass α (fun a => C (key b) a ∧ pinFailure (V a) (imageB b).Y (S b))) ≤
    48 η*(∑ b,β b*cellMass α (C (key b))) + τ +
    49 Fintype.card I*(((2*MB)*((2 : ℝ)^(Fintype.card H*K+2*K*ℓ)/(2 : ℝ)^(Fintype.card H*ℓ)))/
    50 (η-(M/τ)*(2*((2 : ℝ)^(Fintype.card P+Fintype.card H+K*ℓ)/(2 : ℝ)^Fintype.card N)))^ℓ)
    51
    52axiom exponential_accepting_injection {A B I P N H : Type}
    53 [Fintype A] [Fintype B] [Fintype I] [Fintype P] [Fintype N] [Fintype H]
    54 [DecidableEq P] [DecidableEq N] [DecidableEq H]
    55 (E : Matrix P P Binary) [Nonempty (Frame P H N E)]
    56 (α : A → ℝ) (β : B → ℝ) (C : I → A → Prop) (key : B → I)
    57 (imageA : A → Frame P H N E) (imageB : B → Frame P H N E)
    58 (V : A → Submodule Binary (N → Binary)) (S : B → Submodule Binary (H → Binary))
    59 (K m t Cs : ℕ) (M MB η : ℝ)
    60 (hα : Probability α) (hβ : Probability β) (hM : 0 ≤ M) (hMB : 0 ≤ MB)
    61 (hη : 0 ≤ η) (hηt : 1/(2 : ℝ)^t ≤ η/2)
    62 (hN : Fintype.card P+Fintype.card H+1 ≤ Fintype.card N)
    63 (hb : Fintype.card P ≤ Fintype.card N/8)
    64 (hm : m+Fintype.card H+1 ≤ Fintype.card N/10)
    65 (hMcap : M ≤ (2 : ℝ)^m) (hMBcap : 2*MB ≤ (2 : ℝ)^(m+Fintype.card N))
    66 (hcount : (Fintype.card I : ℝ) ≤ (2 : ℝ)^(Cs*Fintype.card N))
    67 (hearly : 2*K+t+20*(K+1)*(Cs+3) ≤ Fintype.card H)
    68 (hlength : 20*(K+1) ≤ Fintype.card N)
    69 (hconstant : m+Fintype.card H*K ≤ Fintype.card N)
    70 (hsmall : 1/(2 : ℝ)^(Fintype.card N/2) ≤ η/2)
    71 (hV : ∀ a,V a ≤ LinearMap.range (imageA a).P.mulVecLin)
    72 (hdV : ∀ a,Module.finrank Binary (V a) ≤ K)
    73 (hS : ∀ b,Module.finrank Binary (S b) ≤ K)
    74 (hAcap : ∀ f,push α imageA f ≤ M*weights (PMF.uniformOfFintype (Frame P H N E)) f)
    75 (hBcap : ∀ f,push β imageB f ≤ MB*weights (PMF.uniformOfFintype (Frame P H N E)) f) :
    76 (∑ b,β b*cellMass α (fun a => C (key b) a ∧ pinFailure (V a) (imageB b).Y (S b))) ≤
    77 η*(∑ b,β b*cellMass α (C (key b))) + 1/(2 : ℝ)^(Fintype.card N/100) + 1/(2 : ℝ)^Fintype.card N
    78
    79end Lax342547.RawAcceptingInjection
    80
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