Lax Lean Archive

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

Actual finite accepting-pair injection

Lax342547.AcceptingInjection · concepts/Lax342547/AcceptingInjection.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

    Greedy pin probes and whole-unit protected spaces give a finite accepting-pair loss bound for the entire accepting family, with raw primal avoidance and retained masses.

    Concept map
    90 concepts; 4 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 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 spacesExact probabilities for independent protectedchannel imagesWalsh 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 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 2 statements. Each proof establishes one of them relative to its assumptions.

    1 accepting_family_injection proven

    2 primal_accepting_family_injection proven

    Lean source view on GitHub

    1import Lax342547.FiniteInjection
    2import Lax342547.AcceptingFamilies
    3import Lax342547.UnitSpanAvoidance
    4import Lax342547.GreedySpaces
    5
    6/-!
    7---
    8title: Actual finite accepting-pair injection
    9type: lemma
    10---
    11Greedy pin probes and whole-unit protected spaces give a finite accepting-pair loss bound for the entire accepting family, with raw primal avoidance and retained masses.
    12-/
    13
    14namespace Lax342547.AcceptingInjection
    15
    16open Lax342547.MomentSpace Lax342547.RelativeEntropy Lax342547.RetainedImages
    17open Lax342547.RealCellLaws Lax342547.PushforwardWalsh Lax342547.FiniteInjection
    18open Lax342547.RawFrames
    19open scoped BigOperators
    20
    21axiom accepting_family_injection {A B I N H : Type}
    22 [Fintype A] [Fintype B] [Fintype I] [Fintype N] [Fintype H]
    23 [DecidableEq N] [DecidableEq H]
    24 (α : A → ℝ) (β : B → ℝ) (C : I → A → Prop) (key : B → I)
    25 (V : A → Submodule Binary (N → Binary)) (Y : B → Matrix N H Binary)
    26 (S : B → Submodule Binary (H → Binary)) (K ℓ : ℕ) (L η δ τ : ℝ)
    27 (hα : Probability α) (hβ : Probability β) (hL : 0 ≤ L) (hη : 0 ≤ η)
    28 (hτ : 0 < τ) (hgap : δ < η)
    29 (hV : ∀ a,Module.finrank Binary (V a) ≤ K)
    30 (hS : ∀ b,Module.finrank Binary (S b) ≤ K)
    31 (hcap : ∀ y,push β Y y ≤ L*weights (PMF.uniformOfFintype (Matrix N H Binary)) y)
    32 (havoid : ∀ i,τ ≤ cellMass α (C i) → ∀ j,j < ℓ → ∀ t : Fin j → A,
    33 cellMass (conditionalLaw α (C i)) (fun a => ¬ Disjoint (V a) (⨆ k : Fin j,V (t k))) ≤ δ) :
    34 (∑ b,β b*cellMass α (fun a => C (key b) a ∧ pinFailure (V a) (Y b) (S b))) ≤
    35 η*(∑ b,β b*cellMass α (C (key b))) + τ +
    36 Fintype.card I*((L*((2 : ℝ)^(Fintype.card H*K+2*K*ℓ)/(2 : ℝ)^(Fintype.card H*ℓ)))/(η-δ)^ℓ)
    37
    38axiom primal_accepting_family_injection {A B I P N H : Type}
    39 [Fintype A] [Fintype B] [Fintype I] [Fintype P] [Fintype N] [Fintype H]
    40 [DecidableEq P] [DecidableEq N] [DecidableEq H]
    41 (E : Matrix P P Binary) [Nonempty (Frame P H N E)]
    42 (α : A → ℝ) (β : B → ℝ) (C : I → A → Prop) (key : B → I)
    43 (image : A → Frame P H N E) (V : A → Submodule Binary (N → Binary))
    44 (Y : B → Matrix N H Binary) (S : B → Submodule Binary (H → Binary))
    45 (K ℓ : ℕ) (M L η τ : ℝ)
    46 (hα : Probability α) (hβ : Probability β) (hM : 0 ≤ M) (hL : 0 ≤ L)
    47 (hη : 0 ≤ η) (hτ : 0 < τ)
    48 (hN : Fintype.card P+Fintype.card H+1 ≤ Fintype.card N)
    49 (hV : ∀ a,V a ≤ LinearMap.range (image a).P.mulVecLin)
    50 (hdV : ∀ a,Module.finrank Binary (V a) ≤ K)
    51 (hS : ∀ b,Module.finrank Binary (S b) ≤ K)
    52 (hAcap : ∀ f,push α image f ≤ M*weights (PMF.uniformOfFintype (Frame P H N E)) f)
    53 (hBcap : ∀ y,push β Y y ≤ L*weights (PMF.uniformOfFintype (Matrix N H Binary)) y)
    54 (hgap : (M/τ)*(2*((2 : ℝ)^(Fintype.card P+Fintype.card H+K*ℓ)/(2 : ℝ)^Fintype.card N)) < η) :
    55 (∑ b,β b*cellMass α (fun a => C (key b) a ∧ pinFailure (V a) (Y b) (S b))) ≤
    56 η*(∑ b,β b*cellMass α (C (key b))) + τ +
    57 Fintype.card I*((L*((2 : ℝ)^(Fintype.card H*K+2*K*ℓ)/(2 : ℝ)^(Fintype.card H*ℓ)))/
    58 (η-(M/τ)*(2*((2 : ℝ)^(Fintype.card P+Fintype.card H+K*ℓ)/(2 : ℝ)^Fintype.card N)))^ℓ)
    59
    60end Lax342547.AcceptingInjection
    61
    Show 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…