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Uniform mixing forms chosen simultaneously before any unit law

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

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    Natural Language Statement

    Lemma

    Lemma 5.5: one choice of all indexed L,R matrices and all sharp M matrices has the unary polar-rank and bounded-row-space properties and every binary parity rank bound. An explicit threshold in n pays for both finite unions. The paper's displayed mixer parameters satisfy the required margin.

    Concept map
    28 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

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

    1 exists_paper_mixers proven

    2 exists_uniform_mixers proven

    3 paper_parameters proven

    Lean source view on GitHub

    1import Lax342547.UnaryMixers
    2import Lax342547.BinaryMixers
    3
    4/-!
    5---
    6title: Uniform mixing forms chosen simultaneously before any unit law
    7type: lemma
    8---
    9Lemma 5.5: one choice of all indexed L,R matrices and all sharp M matrices
    10has the unary polar-rank and bounded-row-space properties and every binary
    11parity rank bound. An explicit threshold in n pays for both finite unions.
    12The paper's displayed mixer parameters satisfy the required margin.
    13-/
    14
    15namespace Lax342547.UniformMixers
    16
    17open Lax342547.MomentSpace Lax342547.TagGeometry Lax342547.ConcreteGeometry
    18open Lax342547.ConcreteCut Lax342547.CutProfiles Lax342547.UnaryRowLaw
    19
    20noncomputable def threshold (k b degree J K s₀ : ℕ) : ℕ :=
    21 max 50 (UnaryMixers.threshold k b degree J K s₀ +
    22 2 ^ (2 * b + 2 * Fintype.card (Index (Component (Tag k)) J)) + 2 ^ (2 * b + 2))
    23
    24def paperS₀ (k b degree K : ℕ) : ℕ :=
    25 40 * (K + 1) * (∑ j ∈ Finset.range (degree + 1), b.choose j) * ((2 * k + 1) ^ 2 + 5)
    26
    27def paperJ (k b degree K : ℕ) : ℕ := 101 * (paperS₀ k b degree K + K ^ 2 + 1)
    28
    29axiom exists_uniform_mixers (k b degree J K s₀ : ℕ) (hk : 0 < k) (hd : 1 ≤ degree)
    30 (hmargin : (∑ j ∈ Finset.range (degree + 1), b.choose j) * ((2 * k + 1) ^ 2 + 3) * K ≤ s₀)
    31 (n : ℕ) (hn : threshold k b degree J K s₀ ≤ n)
    32 (r₀ : ℕ) (hr : 2 * r₀ ≤ n) (D : Testers (k := k) (b := b) (degree := degree) hr) :
    33 ∃ L : Index (Component (Tag k)) J → Moment k n b degree,
    34 ∃ M : Fin b → Matrix (Fin n) (Fin n) Binary,
    35 UnaryMixers.UniformUnary D K s₀ L ∧ BinaryMixers.UniformBinary hr hd L M
    36
    37axiom paper_parameters (k b degree K : ℕ) :
    38 (∑ j ∈ Finset.range (degree + 1), b.choose j) * ((2 * k + 1) ^ 2 + 3) * K ≤ paperS₀ k b degree K ∧
    39 100 * (paperS₀ k b degree K + K ^ 2 + 1) < paperJ k b degree K
    40
    41axiom exists_paper_mixers (k b degree K : ℕ) (hk : 0 < k) (hd : 1 ≤ degree) :
    42 ∃ n₀, ∀ n, n₀ ≤ n → ∀ r₀ (hr : 2 * r₀ ≤ n)
    43 (D : Testers (k := k) (b := b) (degree := degree) hr),
    44 ∃ L : Index (Component (Tag k)) (paperJ k b degree K) → Moment k n b degree,
    45 ∃ M : Fin b → Matrix (Fin n) (Fin n) Binary,
    46 UnaryMixers.UniformUnary D K (paperS₀ k b degree K) L ∧ BinaryMixers.UniformBinary hr hd L M
    47
    48end Lax342547.UniformMixers
    49
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