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Trimming a restricted family of exact-pin leaves

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

proven

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    Natural Language Statement

    Theorem

    Retained fractions are weighted by the actual original leaf masses. A bound on their combined cost suffices to keep every surviving leaf's image and density estimates, with a uniform explicit discarded fraction.

    Concept map
    9 concepts
    100%
    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 restricted_family proven

    2 weighted_residual proven

    Lean source view on GitHub

    1import Lax342547.LeafTrimming
    2import Lax342547.BoundedPeeling
    3
    4/-!
    5---
    6title: Trimming a restricted family of exact-pin leaves
    7type: theorem
    8---
    9Retained fractions are weighted by the actual original leaf masses.
    10A bound on their combined cost suffices to keep every surviving leaf's
    11image and density estimates, with a uniform explicit discarded fraction.
    12-/
    13
    14namespace Lax342547.RestrictedLeaves
    15
    16open Lax342547.MomentSpace Lax342547.ExactPins
    17open Lax342547.LeafTrimming Lax342547.BoundedPeeling
    18open scoped ENNReal
    19
    20axiom restricted_family {Ω J Axis I N : Type}
    21 [Fintype Ω] [Fintype J] [Fintype Axis]
    22 (p : J → PMF Ω) (μ : PMF Ω)
    23 (A : Ω → Axis → (I → Binary) →ₗ[Binary] (N → Binary))
    24 (P : J → Pin Axis I N) (S : J → Set Ω) (w : J → ℝ≥0∞)
    25 (C ζ η : ℝ) (n : ℕ) (hζ : 0 ≤ ζ) (hw : ∑ j, w j ≤ 1)
    26 (hq : (2 : ℝ≥0∞) ^ (-(η * n)) ≤ ∑ j, w j * (p j).toOuterMeasure (S j))
    27 (hcap : ∀ j (Q : Pin Axis I N), 1 ≤ (P j).relativeRank Q →
    28 (p j).toOuterMeasure (Q.event A) ≤
    29 ((2 : ℝ≥0∞) ^ (-((1 - ζ) * n))) ^ (P j).relativeRank Q)
    30 (hdensity : ∀ j o, p j o ≤ (2 : ℝ≥0∞) ^ (C * n) * μ o) :
    31 discardedMass w (fun j => (p j).toOuterMeasure (S j)) ((2 : ℝ≥0∞) ^ (-(ζ * n))) /
    32 (∑ j, w j * (p j).toOuterMeasure (S j)) ≤ (2 : ℝ≥0∞) ^ (-((ζ - η) * n)) ∧
    33 ∀ j, (2 : ℝ≥0∞) ^ (-(ζ * n)) ≤ (p j).toOuterMeasure (S j) →
    34 LawBound (p j) μ A (P j) (S j)
    35 ((2 : ℝ≥0∞) ^ (-((1 - 2 * ζ) * n))) ((2 : ℝ≥0∞) ^ ((C + ζ) * n))
    36
    37axiom weighted_residual {J : Type} [Fintype J] (w r : J → ℝ≥0∞) (δ : ℝ≥0∞)
    38 (hw : ∑ j, w j ≤ 1) (hr : ∀ j, r j ≤ δ) : ∑ j, w j * r j ≤ δ
    39
    40end Lax342547.RestrictedLeaves
    41
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