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The finite uniform raw-vertex law

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

proven

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    Natural Language Statement

    Lemma

    A raw vertex chooses one valid frame in every component. The reference law is uniform on this finite product. Counting all matrix quadruples gives ∣Ω∣≤22∣E∣N(d+h)|\Omega|\le 2^{2|\mathcal E|N(d+h)}, the bound behind (2.14).

    Concept map
    3 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 raw_card_bound proven

    2 uniformLaw_product proven

    Lean source view on GitHub

    1import Lax342547.RawFrames
    2import Mathlib.Probability.Distributions.Uniform
    3
    4/-!
    5---
    6title: The finite uniform raw-vertex law
    7type: lemma
    8---
    9A raw vertex chooses one valid frame in every component. The reference
    10law is uniform on this finite product. Counting all matrix quadruples
    11gives ∣Ω∣≤22∣E∣N(d+h)|\Omega|\le 2^{2|\mathcal E|N(d+h)}, the bound behind (2.14).
    12-/
    13
    14namespace Lax342547.RawLaw
    15
    16open Lax342547.MomentSpace Lax342547.RawFrames
    17
    18variable {B H N Comp : Type} [Fintype B] [Fintype H] [Fintype N] [Fintype Comp]
    19variable [DecidableEq Comp]
    20
    21noncomputable def uniformLaw (E : Matrix B B Binary) [Nonempty (Frame B H N E)] :
    22 PMF (Comp → Frame B H N E) := PMF.uniformOfFintype _
    23
    24axiom raw_card_bound (E : Matrix B B Binary) :
    25 Fintype.card (Comp → Frame B H N E) ≤
    26 2 ^ (2 * Fintype.card Comp * Fintype.card N * (Fintype.card B + Fintype.card H))
    27
    28axiom uniformLaw_product (E : Matrix B B Binary) [Nonempty (Frame B H N E)]
    29 (o : Comp → Frame B H N E) :
    30 uniformLaw E o = ∏ e, PMF.uniformOfFintype (Frame B H N E) (o e)
    31
    32end Lax342547.RawLaw
    33
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