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Squared restriction cost for independent unit events

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

proven

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    Natural Language Statement

    Lemma

    A normalized restriction is dominated pointwise by its original PMF after multiplication by the retained mass. Independent pair events therefore lose precisely the square of that mass.

    Concept map
    7 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 6 statements. Each proof establishes one of them relative to its assumptions.

    1 independent_pair_apply proven

    2 pair_mass_domination proven

    3 pair_mass_pmf_event proven

    4 real_filter_domination proven

    5 restriction_pair_event_recovery proven

    6 restriction_pair_recovery proven

    Lean source view on GitHub

    1import Lax342547.RealCellLaws
    2import Lax342547.Conditioning
    3
    4/-!
    5---
    6title: Squared restriction cost for independent unit events
    7type: lemma
    8---
    9A normalized restriction is dominated pointwise by its original PMF after multiplication by the retained mass. Independent pair events therefore lose precisely the square of that mass.
    10-/
    11
    12namespace Lax342547.PairRecovery
    13
    14open Lax342547.RealCellLaws
    15open scoped ENNReal
    16
    17noncomputable def pairMass {ΩA ΩB : Type} [Fintype ΩA] [Fintype ΩB]
    18 (α : ΩA → ℝ) (β : ΩB → ℝ) (H : ΩA → ΩB → Prop) : ℝ := by
    19 classical
    20 exact ∑ a, ∑ b, if H a b then α a * β b else 0
    21
    22axiom pair_mass_domination {ΩA ΩB : Type} [Fintype ΩA] [Fintype ΩB]
    23 (α ρ : ΩA → ℝ) (β σ : ΩB → ℝ) (mA mB : ℝ) (H : ΩA → ΩB → Prop)
    24 (hmA : 0 ≤ mA) (hmB : 0 ≤ mB) (hρ : ∀ a, 0 ≤ ρ a) (hσ : ∀ b, 0 ≤ σ b)
    25 (hα : ∀ a, mA * ρ a ≤ α a) (hβ : ∀ b, mB * σ b ≤ β b) :
    26 mA * mB * pairMass ρ σ H ≤ pairMass α β H
    27
    28axiom real_filter_domination {Ω : Type} [Fintype Ω] (p : PMF Ω) (S : Set Ω)
    29 (hS : ∃ o ∈ S, o ∈ p.support) (ω : Ω) :
    30 (p.toOuterMeasure S).toReal * weights (p.filter S hS) ω ≤ weights p ω
    31
    32axiom restriction_pair_recovery {Ω : Type} [Fintype Ω] (p : PMF Ω) (S : Set Ω)
    33 (hS : ∃ o ∈ S, o ∈ p.support) (H : Ω → Ω → Prop) :
    34 ((p.toOuterMeasure S).toReal)^2 * pairMass (weights (p.filter S hS)) (weights (p.filter S hS)) H ≤
    35 pairMass (weights p) (weights p) H
    36
    37noncomputable def independentPair {ΩA ΩB : Type} (p : PMF ΩA) (q : PMF ΩB) : PMF (ΩA × ΩB) :=
    38 p.bind (fun a => q.map (fun b => (a,b)))
    39
    40axiom independent_pair_apply {ΩA ΩB : Type} [Fintype ΩA] [Fintype ΩB]
    41 (p : PMF ΩA) (q : PMF ΩB) (ab : ΩA × ΩB) :
    42 independentPair p q ab = p ab.1 * q ab.2
    43
    44axiom pair_mass_pmf_event {ΩA ΩB : Type} [Fintype ΩA] [Fintype ΩB]
    45 (p : PMF ΩA) (q : PMF ΩB) (H : ΩA → ΩB → Prop) :
    46 pairMass (weights p) (weights q) H =
    47 ((independentPair p q).toOuterMeasure {ab | H ab.1 ab.2}).toReal
    48
    49axiom restriction_pair_event_recovery {Ω : Type} [Fintype Ω]
    50 (p : PMF Ω) (S : Set Ω) (hS : ∃ o ∈ S, o ∈ p.support) (H : Ω → Ω → Prop) :
    51 ((p.toOuterMeasure S).toReal)^2 *
    52 ((independentPair (p.filter S hS) (p.filter S hS)).toOuterMeasure {ab | H ab.1 ab.2}).toReal ≤
    53 ((independentPair p p).toOuterMeasure {ab | H ab.1 ab.2}).toReal
    54
    55end Lax342547.PairRecovery
    56
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