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Mass and density costs of splitting by metadata and images

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

proven

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    Natural Language Statement

    Theorem

    Discarding small fibers costs at most the number of split records times the threshold. The remaining fibers have an absolute joint density cap suitable for a fresh peeling, even after a preceding leaf restriction.

    Concept map
    6 concepts; 2 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 4 statements. Each proof establishes one of them relative to its assumptions.

    1 exponential_discard proven

    2 metadata_discard proven

    3 repinning_density proven

    4 small_fibers proven

    Lean source view on GitHub

    1import Lax342547.LeafTrimming
    2import Lax342547.MetadataPins
    3
    4/-!
    5---
    6title: Mass and density costs of splitting by metadata and images
    7type: theorem
    8---
    9Discarding small fibers costs at most the number of split records times
    10the threshold. The remaining fibers have an absolute joint density cap
    11suitable for a fresh peeling, even after a preceding leaf restriction.
    12-/
    13
    14namespace Lax342547.SplitTrimming
    15
    16open Lax342547.MomentSpace Lax342547.MetadataPins
    17open scoped ENNReal
    18
    19axiom small_fibers {Ω K : Type} [Fintype K] (p : PMF Ω) (f : Ω → K) (τ : ℝ≥0∞) :
    20 p.toOuterMeasure {o | p.toOuterMeasure {x | f x = f o} < τ} ≤ Fintype.card K * τ
    21
    22axiom metadata_discard {Ω J Axis I N : Type}
    23 [Fintype J] [Fintype Axis] [Fintype I] [Fintype N]
    24 (p : PMF Ω) (A : Ω → Axis → (I → Binary) →ₗ[Binary] (N → Binary))
    25 (choice : Ω → J) (S : J → Axis → Submodule Binary (I → Binary)) (a n u : ℕ)
    26 (hchoices : Fintype.card J ≤ 2 ^ (a * n))
    27 (hu : ∀ j, ∑ b, Module.finrank Binary (S j b) ≤ u) :
    28 p.toOuterMeasure {o | p.toOuterMeasure {x | record A choice S x = record A choice S o} <
    29 (2 : ℝ≥0∞) ^ (-(((u : ℝ) + 1) * Fintype.card N))} ≤
    30 (2 : ℝ≥0∞) ^ ((a * n : ℕ) - (Fintype.card N : ℝ))
    31
    32axiom exponential_discard {Ω J Axis I N : Type}
    33 [Fintype J] [Fintype Axis] [Fintype I] [Fintype N]
    34 (p : PMF Ω) (A : Ω → Axis → (I → Binary) →ₗ[Binary] (N → Binary))
    35 (choice : Ω → J) (S : J → Axis → Submodule Binary (I → Binary)) (a n u : ℕ)
    36 (hchoices : Fintype.card J ≤ 2 ^ (a * n))
    37 (hu : ∀ j, ∑ b, Module.finrank Binary (S j b) ≤ u)
    38 (hscale : 2 * a * n ≤ Fintype.card N) :
    39 p.toOuterMeasure {o | p.toOuterMeasure {x | record A choice S x = record A choice S o} <
    40 (2 : ℝ≥0∞) ^ (-(((u : ℝ) + 1) * Fintype.card N))} ≤
    41 (2 : ℝ≥0∞) ^ (-(Fintype.card N : ℝ) / 2)
    42
    43axiom repinning_density {Ω : Type} (p μ : PMF Ω) (S T : Set Ω)
    44 (hS : ∃ o ∈ S, o ∈ p.support)
    45 (hT : ∃ o ∈ T, o ∈ (p.filter S hS).support)
    46 (D ζ : ℝ) (N K₁ u : ℕ) (hζ : ζ ≤ 1)
    47 (hSbound : (2 : ℝ≥0∞) ^ (-(ζ * N)) ≤ p.toOuterMeasure S)
    48 (hTbound : (2 : ℝ≥0∞) ^ (-(((u : ℝ) + 1) * N)) ≤ (p.filter S hS).toOuterMeasure T)
    49 (hp : ∀ o, p o ≤ (2 : ℝ≥0∞) ^ ((D + K₁ + 1) * N) * μ o) :
    50 ∀ o, ((p.filter S hS).filter T hT) o ≤
    51 (2 : ℝ≥0∞) ^ ((D + K₁ + u + 5) * N) * μ o
    52
    53end Lax342547.SplitTrimming
    54
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