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The full reference cap for exact pin events

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

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    Natural Language Statement

    Lemma

    Component/sign pins use the direct sum of both endpoints' primal and channel coefficients. Their stored basis images form precisely the arbitrary-rank tuples covered by the reference image theorem.

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

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax342547.ExactPins
    2import Lax342547.ImageScale
    3
    4/-!
    5---
    6title: The full reference cap for exact pin events
    7type: lemma
    8---
    9Component/sign pins use the direct sum of both endpoints' primal and
    10channel coefficients. Their stored basis images form precisely the
    11arbitrary-rank tuples covered by the reference image theorem.
    12-/
    13
    14namespace Lax342547.ReferencePins
    15
    16open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.FrameSymmetry
    17open Lax342547.ProductImages Lax342547.ExactPins
    18open scoped ENNReal
    19
    20variable {Comp B H N : Type} [Fintype B] [Fintype H] [Fintype N]
    21
    22def observationMatrix {E : Matrix B B Binary} (o : Fin 2 → Comp → Frame B H N E)
    23 (a : Comp × Bool) : Matrix N (Fin 2 × (B ⊕ H)) Binary :=
    24 if a.2 then jointMatrix (fun i => (plus (o i a.1)).val)
    25 else jointMatrix (fun i => (minus (o i a.1)).val)
    26
    27def observation {E : Matrix B B Binary} (o : Fin 2 → Comp → Frame B H N E)
    28 (a : Comp × Bool) : ((Fin 2 × (B ⊕ H)) → Binary) →ₗ[Binary] (N → Binary) :=
    29 (observationMatrix o a).mulVecLin
    30
    31axiom reference_pin_cap [Fintype Comp]
    32 [DecidableEq Comp] [DecidableEq B] [DecidableEq H] [DecidableEq N]
    33 {E : Matrix B B Binary} [Nonempty (Frame B H N E)]
    34 (ε : ℝ) (hε : 0 < ε)
    35 (hN : 2 * (Fintype.card B + Fintype.card H) + 1 ≤ Fintype.card N)
    36 (hp : (4 : ℝ) * (Fintype.card B + Fintype.card H : ℕ) ≤ ε * Fintype.card N)
    37 (hc : (8 : ℝ) * Fintype.card Comp ≤ ε * Fintype.card N)
    38 (P : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N) :
    39 (PMF.uniformOfFintype (Fin 2 → Comp → Frame B H N E)).toOuterMeasure (P.event observation) ≤
    40 (2 : ℝ≥0∞) ^ (-((1 - ε) * P.rank * Fintype.card N))
    41
    42end Lax342547.ReferencePins
    43
    Show Proof
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    From Mathlib

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