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Exact finite conditioning of separated Fourier tests

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

proven

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    Natural Language Statement

    Lemma

    Fourier inversion controls signed test integrals and their normalization separately. Division by the actual conditioning mass gives a quantitative comparison, uniform over bounded tests.

    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 3 statements. Each proof establishes one of them relative to its assumptions.

    1 bounded_integral_deletion proven

    2 conditional_test_comparison proven

    3 signed_pattern_error proven

    Lean source view on GitHub

    1import Lax342547.FourierTests
    2import Lax342547.RelativeEntropy
    3
    4/-!
    5---
    6title: Exact finite conditioning of separated Fourier tests
    7type: lemma
    8---
    9Fourier inversion controls signed test integrals and their normalization
    10separately. Division by the actual conditioning mass gives a quantitative
    11comparison, uniform over bounded tests.
    12-/
    13
    14namespace Lax342547.ConditionedMixing
    15
    16open Lax342547.MomentSpace Lax342547.Walsh Lax342547.FourierTests
    17open scoped BigOperators
    18
    19axiom signed_pattern_error {Ω S : Type} [Fintype Ω] [Fintype S] [DecidableEq S]
    20 (w : Ω → ℝ) (bits : Ω → S → Binary) (target : S → Binary) (ε : ℝ)
    21 (hε : 0 ≤ ε) (hchar : ∀ t : S → Binary,t ≠ 0 → |characterMean w bits target t| ≤ ε) :
    22 |patternMass w bits target-(∑ ω,w ω)/(2 : ℝ)^Fintype.card S| ≤ ε
    23
    24axiom conditional_test_comparison {Ω S : Type} [Fintype Ω] [Fintype S] [DecidableEq S]
    25 (ρ f : Ω → ℝ) (bits : Ω → S → Binary) (target : S → Binary) (ε : ℝ)
    26 (hε : 0 ≤ ε) (htotal : ∑ ω,ρ ω = 1) (hmean : |∑ ω,ρ ω*f ω| ≤ 1)
    27 (hraw : ∀ t : S → Binary,t ≠ 0 → |characterMean ρ bits target t| ≤ ε)
    28 (htest : ∀ t : S → Binary,t ≠ 0 → |characterMean (fun ω => ρ ω*f ω) bits target t| ≤ ε)
    29 (hmass : 0 < patternMass ρ bits target) :
    30 |patternMass (fun ω => ρ ω*f ω) bits target/patternMass ρ bits target-
    31 (∑ ω,ρ ω*f ω)| ≤ 2*ε/patternMass ρ bits target
    32
    33axiom bounded_integral_deletion {Ω : Type} [Fintype Ω] (ρ f : Ω → ℝ)
    34 (C : Ω → Prop) (hρ : Lax342547.RelativeEntropy.Probability ρ)
    35 (hf : ∀ x,|f x| ≤ 1) (hC : 0 < Lax342547.RetainedImages.cellMass ρ C) : by
    36 classical
    37 exact |(∑ x,if C x then ρ x*f x else 0)/Lax342547.RetainedImages.cellMass ρ C-
    38 (∑ x,ρ x*f x)| ≤ 2*(1-Lax342547.RetainedImages.cellMass ρ C)
    39
    40end Lax342547.ConditionedMixing
    41
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