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Union over adaptive component covers

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

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    Natural Language Statement

    Lemma

    All disjoint exposed and tested index sets have the finite union bound four to the number of coordinates times the cover mass to the number of tests.

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

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax342547.AdaptiveSets
    2
    3/-!
    4---
    5title: Union over adaptive component covers
    6type: lemma
    7---
    8All disjoint exposed and tested index sets have the finite union bound four to the number of coordinates times the cover mass to the number of tests.
    9-/
    10
    11namespace Lax342547.AdaptiveUnion
    12
    13open Lax342547.RelativeEntropy Lax342547.FiniteSampling Lax342547.RetainedImages
    14open scoped BigOperators
    15
    16axiom adaptive_cover_union {ι Ω : Type} [Fintype ι] [Fintype Ω] [DecidableEq ι]
    17 (μ : Ω → ℝ) (A : ∀ E : Finset ι, (E → Ω) → Ω → Prop) (p : ℝ) (k : ℕ)
    18 (hμ : Probability μ) (hp0 : 0 ≤ p) (hp1 : p ≤ 1)
    19 (hA : ∀ E exposed, cellMass μ (A E exposed) ≤ p) :
    20 cellMass (productLaw (fun _ : ι => μ))
    21 (fun sample => ∃ E J : Finset ι, Disjoint E J ∧ k ≤ J.card ∧
    22 ∀ j ∈ J, A E (fun e => sample e.val) (sample j)) ≤
    23 (4 : ℝ)^Fintype.card ι*p^k
    24
    25end Lax342547.AdaptiveUnion
    26
    Show Proof
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    From Mathlib

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