Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Finite even moment expansion

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

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Lemma

    An actual finite even moment expands over independent samples. Bounded unit-side signs may be removed whenever the channel character expectation is nonnegative.

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

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 finite_even_moment proven

    2 weighted_character_moment_bound proven

    Lean source view on GitHub

    1import Lax342547.FiniteSampling
    2import Mathlib.Algebra.Order.BigOperators.Group.Finset
    3
    4/-!
    5---
    6title: Finite even moment expansion
    7type: lemma
    8---
    9An actual finite even moment expands over independent samples. Bounded unit-side signs may be removed whenever the channel character expectation is nonnegative.
    10-/
    11
    12namespace Lax342547.FiniteMoments
    13
    14open Lax342547.RelativeEntropy Lax342547.FiniteSampling Lax342547.RetainedImages
    15open scoped BigOperators
    16
    17axiom finite_even_moment {Ω C : Type} [Fintype Ω] [Fintype C]
    18 (μ : Ω → ℝ) (ν : C → ℝ) (ψ : Ω → ℝ) (χ : Ω → C → ℝ) (t : ℕ) :
    19 (∑ c, ν c*|∑ x, μ x*ψ x*χ x c|^(2*t)) =
    20 ∑ sample : Fin (2*t) → Ω, productLaw (fun _ : Fin (2*t) => μ) sample*
    21 (∏ i, ψ (sample i))*(∑ c, ν c*∏ i, χ (sample i) c)
    22
    23axiom weighted_character_moment_bound {Ω C : Type} [Fintype Ω] [Fintype C]
    24 (μ : Ω → ℝ) (ν : C → ℝ) (ψ : Ω → ℝ) (χ : Ω → C → ℝ) (t : ℕ)
    25 (hμ : ∀ x, 0 ≤ μ x) (hψ : ∀ x, |ψ x| ≤ 1)
    26 (hχ : ∀ sample : Fin (2*t) → Ω, 0 ≤ ∑ c, ν c*∏ i, χ (sample i) c) :
    27 (∑ c, ν c*|∑ x, μ x*ψ x*χ x c|^(2*t)) ≤
    28 ∑ sample : Fin (2*t) → Ω, productLaw (fun _ : Fin (2*t) => μ) sample*
    29 (∑ c, ν c*∏ i, χ (sample i) c)
    30
    31end Lax342547.FiniteMoments
    32
    Show ProofShow Proof
    Builds on
    Used by
    From Mathlib

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…