Lax Lean Archive

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

Scalar agreement after quotient character estimates

Lax342547.GramTests · concepts/Lax342547/GramTests.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

    The .95 image exponent gives the .45 Walsh exponent. Exact zero quotient characters and bounded nonzero characters give simultaneous scalar agreement with the full binary Fourier cost.

    Concept map
    6 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 paper_image_walsh proven

    2 paper_walsh_factor proven

    3 zero_or_biased_agreement proven

    Lean source view on GitHub

    1import Lax342547.RetainedImages
    2import Lax342547.FourierTests
    3
    4/-!
    5---
    6title: Scalar agreement after quotient character estimates
    7type: lemma
    8---
    9The .95 image exponent gives the .45 Walsh exponent. Exact zero quotient characters and bounded nonzero characters give simultaneous scalar agreement with the full binary Fourier cost.
    10-/
    11
    12namespace Lax342547.GramTests
    13
    14open Lax342547.MomentSpace Lax342547.Walsh Lax342547.PushforwardWalsh
    15open Lax342547.FourierTests
    16
    17axiom paper_walsh_factor (d : ℝ) :
    18 Real.sqrt ((2 : ℝ)^d * (2 : ℝ)^(-(95/100 : ℝ)*d) * (2 : ℝ)^(-(95/100 : ℝ)*d)) =
    19 (2 : ℝ)^(-(45/100 : ℝ)*d)
    20
    21axiom paper_image_walsh {ΩA ΩB I : Type} [Fintype ΩA] [Fintype ΩB]
    22 [Fintype I] [DecidableEq I]
    23 (α : ΩA → ℝ) (β : ΩB → ℝ) (imageA : ΩA → I → Binary) (imageB : ΩB → I → Binary)
    24 (f : ΩA → ℝ) (g : ΩB → ℝ)
    25 (hα : ∀ a, 0 ≤ α a) (hβ : ∀ b, 0 ≤ β b)
    26 (hαsum : ∑ a, α a ≤ 1) (hβsum : ∑ b, β b ≤ 1)
    27 (hcapA : ∀ x, push α imageA x ≤ (2 : ℝ)^(-(95/100 : ℝ)*Fintype.card I))
    28 (hcapB : ∀ y, push β imageB y ≤ (2 : ℝ)^(-(95/100 : ℝ)*Fintype.card I))
    29 (hf : ∀ a, |f a| ≤ 1) (hg : ∀ b, |g b| ≤ 1) :
    30 |∑ a, ∑ b, α a * β b * f a * g b * phase (imageA a) (imageB b)| ≤
    31 (2 : ℝ)^(-(45/100 : ℝ)*Fintype.card I)
    32
    33axiom zero_or_biased_agreement {Ω I : Type} [Fintype Ω] [Fintype I] [DecidableEq I]
    34 (ρ : Ω → ℝ) (bits : Ω → I → Binary) (target : I → Binary) (ε : ℝ)
    35 (hε : 0 ≤ ε) (hρ : ∑ ω, ρ ω = 1)
    36 (hchar : ∀ t : I → Binary, t ≠ 0 →
    37 characterMean ρ bits target t = 1 ∨ |characterMean ρ bits target t| ≤ ε)
    38 (hsmall : ε ≤ 1 / (2 : ℝ) ^ (Fintype.card I + 1)) :
    39 1 / (2 : ℝ) ^ (Fintype.card I + 1) ≤ patternMass ρ bits target
    40
    41end Lax342547.GramTests
    42
    Show ProofShow ProofShow Proof
    Builds on
    Used by
    From Mathlib

    none

    Discussion

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

    Loading discussion…