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Accepting-pair test unions and relative loss

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

proven

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    Natural Language Statement

    Lemma

    Actual product-pair event masses and finite injection tests retain accepting probability; the exponential tail gives arbitrary relative error under inverse-polynomial accepting mass.

    Concept map
    64 concepts; 3 descendants hidden
    100%
    Actual projected span deficitSampling after exposed coordinatesAdaptive sampling on disjoint coordinatesetsUnion over adaptive component coversExact uniform bilinear character meanWalsh operator bounds with explicit bilinearrankRank of a lifted tensor sumIndependent binary channel charactersActual 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 even moment expansionFinite independent sampling and vertexexception tailsGreedy mode space exposureSmall actual greedy exposure tailsNo-cover rank growth on arbitrary finiteindicesRelative injection loss for polynomialaccepting massEarly channel choice and injection exponentbudgetsAccepting-pair test unions and relative lossActual deficits indexed by a distinct listFinite list tail statisticsDimension 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 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 restrictionsOriginal 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 finite_injection_union proven

    2 inverse_polynomial_relative_union proven

    3 pair_event_mass proven

    Lean source view on GitHub

    1import Lax342547.InjectionAsymptotics
    2import Lax342547.FiniteSampling
    3
    4/-!
    5---
    6title: Accepting-pair test unions and relative loss
    7type: lemma
    8---
    9Actual product-pair event masses and finite injection tests retain accepting probability; the exponential tail gives arbitrary relative error under inverse-polynomial accepting mass.
    10-/
    11
    12namespace Lax342547.InjectionUnion
    13
    14open Lax342547.RelativeEntropy Lax342547.RetainedImages
    15open scoped BigOperators
    16
    17axiom pair_event_mass {A B : Type} [Fintype A] [Fintype B] (α : A → ℝ) (β : B → ℝ)
    18 (E : A → B → Prop) :
    19 cellMass (fun p : A × B => α p.1*β p.2) (fun p => E p.1 p.2) =
    20 ∑ b,β b*cellMass α (fun a => E a b)
    21
    22axiom finite_injection_union {A B J : Type} [Fintype A] [Fintype B] [Fintype J]
    23 (α : A → ℝ) (β : B → ℝ) (accept : A → B → Prop) (F : J → A → B → Prop)
    24 (η τ : ℝ) (hα : ∀ a,0 ≤ α a) (hβ : ∀ b,0 ≤ β b)
    25 (hF : ∀ j,(∑ b,β b*cellMass α (fun a => accept a b ∧ F j a b)) ≤
    26 η*(∑ b,β b*cellMass α (fun a => accept a b))+τ) :
    27 cellMass (fun p : A × B => α p.1*β p.2) (fun p => accept p.1 p.2 ∧ ∃ j,F j p.1 p.2) ≤
    28 Fintype.card J*(η*cellMass (fun p : A × B => α p.1*β p.2) (fun p => accept p.1 p.2)+τ)
    29
    30axiom inverse_polynomial_relative_union {A B J : Type} [Fintype A] [Fintype B] [Fintype J]
    31 (α : A → ℝ) (β : B → ℝ) (accept : A → B → Prop) (F : J → A → B → Prop)
    32 (N c : ℕ) (ε : ℝ) (hJ : 0 < Fintype.card J) (hε : 0 ≤ ε)
    33 (hα : ∀ a,0 ≤ α a) (hβ : ∀ b,0 ≤ β b)
    34 (haccept : 1/(N : ℝ)^c ≤ cellMass (fun p : A × B => α p.1*β p.2) (fun p => accept p.1 p.2))
    35 (hF : ∀ j,(∑ b,β b*cellMass α (fun a => accept a b ∧ F j a b)) ≤
    36 (ε/(2*Fintype.card J))*(∑ b,β b*cellMass α (fun a => accept a b))+
    37 (1/(2 : ℝ)^(N/100)+1/(2 : ℝ)^N))
    38 (htail : (Fintype.card J : ℝ)*(1/(2 : ℝ)^(N/100)+1/(2 : ℝ)^N) ≤ (ε/2)/(N : ℝ)^c) :
    39 cellMass (fun p : A × B => α p.1*β p.2) (fun p => accept p.1 p.2 ∧ ∃ j,F j p.1 p.2) ≤
    40 ε*cellMass (fun p : A × B => α p.1*β p.2) (fun p => accept p.1 p.2)
    41
    42end Lax342547.InjectionUnion
    43
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