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Quantitative trimming after restrictions of a leaf mixture

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

proven

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    Natural Language Statement

    Theorem

    Small retained fractions are discarded with their actual mixture weights. The normalized surviving leaves preserve image bounds with an explicit loss, while marginal bounds are tracked on the restricted mixed law.

    Concept map
    2 concepts; 4 descendants hidden
    100%
    Proven claimThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 7 statements. Each proof establishes one of them relative to its assumptions.

    1 combined_restriction proven

    2 density_after_restriction proven

    3 event_after_restriction proven

    4 filtered_marginal proven

    5 minentropy_after_restriction proven

    6 restricted_discard proven

    7 weighted_discard proven

    Lean source view on GitHub

    1import Lax342547.Conditioning
    2import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
    3
    4/-!
    5---
    6title: Quantitative trimming after restrictions of a leaf mixture
    7type: theorem
    8---
    9Small retained fractions are discarded with their actual mixture weights.
    10The normalized surviving leaves preserve image bounds with an explicit
    11loss, while marginal bounds are tracked on the restricted mixed law.
    12-/
    13
    14namespace Lax342547.LeafTrimming
    15
    16open scoped ENNReal
    17
    18noncomputable def discardedMass {J : Type} [Fintype J]
    19 (w q : J → ℝ≥0∞) (δ : ℝ≥0∞) : ℝ≥0∞ :=
    20 ∑ j, if q j < δ then w j * q j else 0
    21
    22axiom weighted_discard {J : Type} [Fintype J] (w q : J → ℝ≥0∞)
    23 (δ : ℝ≥0∞) (hw : ∑ j, w j ≤ 1) : discardedMass w q δ ≤ δ
    24
    25axiom restricted_discard {J : Type} [Fintype J] (w q : J → ℝ≥0∞)
    26 (ζ η : ℝ) (N : ℕ) (hw : ∑ j, w j ≤ 1)
    27 (hq : (2 : ℝ≥0∞) ^ (-(η * N)) ≤ ∑ j, w j * q j) :
    28 discardedMass w q ((2 : ℝ≥0∞) ^ (-(ζ * N))) / (∑ j, w j * q j) ≤
    29 (2 : ℝ≥0∞) ^ (-((ζ - η) * N))
    30
    31axiom event_after_restriction {Ω : Type} [Fintype Ω] (p : PMF Ω)
    32 (S T : Set Ω) (hS : ∃ o ∈ S, o ∈ p.support) (τ b : ℝ≥0∞)
    33 (hτ : τ ≤ p.toOuterMeasure S) (hb : p.toOuterMeasure T ≤ b) :
    34 (p.filter S hS).toOuterMeasure T ≤ b / τ
    35
    36axiom minentropy_after_restriction {Ω : Type} [Fintype Ω] (p : PMF Ω)
    37 (S T : Set Ω) (hS : ∃ o ∈ S, o ∈ p.support) (ζ : ℝ) (N t : ℕ)
    38 (hζ : 0 ≤ ζ) (ht : 1 ≤ t)
    39 (hSbound : (2 : ℝ≥0∞) ^ (-(ζ * N)) ≤ p.toOuterMeasure S)
    40 (hTbound : p.toOuterMeasure T ≤ (2 : ℝ≥0∞) ^ (-((1 - ζ) * t * N))) :
    41 (p.filter S hS).toOuterMeasure T ≤ (2 : ℝ≥0∞) ^ (-((1 - 2 * ζ) * t * N))
    42
    43axiom density_after_restriction {Ω : Type} (p μ : PMF Ω)
    44 (S : Set Ω) (hS : ∃ o ∈ S, o ∈ p.support) (C ζ : ℝ) (N : ℕ)
    45 (hSbound : (2 : ℝ≥0∞) ^ (-(ζ * N)) ≤ p.toOuterMeasure S)
    46 (hp : ∀ o, p o ≤ (2 : ℝ≥0∞) ^ (C * N) * μ o) :
    47 ∀ o, (p.filter S hS) o ≤ (2 : ℝ≥0∞) ^ ((C + ζ) * N) * μ o
    48
    49axiom filtered_marginal {Ω V : Type} [Fintype Ω] (p : PMF Ω) (μ : PMF V)
    50 (f : Ω → V) (S : Set Ω) (hS : ∃ o ∈ S, o ∈ p.support)
    51 (M : ℝ≥0∞) (hp : ∀ v, p.map f v ≤ M * μ v) :
    52 ∀ v, (p.filter S hS).map f v ≤ (M / p.toOuterMeasure S) * μ v
    53
    54axiom combined_restriction {Ω : Type} [Fintype Ω] (p : PMF Ω) (S T : Set Ω)
    55 (hS : ∃ o ∈ S, o ∈ p.support)
    56 (hT : ∃ o ∈ T, o ∈ (p.filter S hS).support)
    57 (hST : ∃ o ∈ S ∩ T, o ∈ p.support) :
    58 (p.filter S hS).filter T hT = p.filter (S ∩ T) hST
    59
    60end Lax342547.LeafTrimming
    61
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