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Exact-pin peeling for the actual raw frame law

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

proven

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    Natural Language Statement

    Theorem

    Apply the finite bounded peeling procedure to both endpoints and both signs of the concrete raw frames. The absolute density bound is always relative to the unconditioned reference law, including when initial pins are retained for a subsequent peeling.

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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax342547.BoundedPeeling
    2import Lax342547.ReferencePins
    3
    4/-!
    5---
    6title: Exact-pin peeling for the actual raw frame law
    7type: theorem
    8---
    9Apply the finite bounded peeling procedure to both endpoints and both
    10signs of the concrete raw frames. The absolute density bound is always
    11relative to the unconditioned reference law, including when initial pins
    12are retained for a subsequent peeling.
    13-/
    14
    15namespace Lax342547.RawPeeling
    16
    17open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.ExactPins
    18open Lax342547.ReferencePins Lax342547.LeafPartition Lax342547.BoundedPeeling
    19open scoped ENNReal
    20
    21axiom raw_peeling {Comp B H N : Type}
    22 [Fintype Comp] [Fintype B] [Fintype H] [Fintype N]
    23 [DecidableEq Comp] [DecidableEq B] [DecidableEq H] [DecidableEq N]
    24 {E : Matrix B B Binary} [Nonempty (Frame B H N E)] [DecidableEq (Frame B H N E)]
    25 (σ : PMF (Fin 2 → Comp → Frame B H N E))
    26 (P₀ : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N)
    27 (R : Finset (Fin 2 → Comp → Frame B H N E))
    28 (hR : (R : Set (Fin 2 → Comp → Frame B H N E)) ⊆ P₀.event observation)
    29 (D ζ ε : ℝ) (K : ℕ)
    30 (hζ : 0 < ζ) (hζ₁ : ζ ≤ 1) (hε : 0 < ε) (hεζ : ε ≤ ζ / 4)
    31 (hK : 4 * (D + 1) / ζ ≤ K)
    32 (hN : 2 * (Fintype.card B + Fintype.card H) + 1 ≤ Fintype.card N)
    33 (hp : (4 : ℝ) * (Fintype.card B + Fintype.card H : ℕ) ≤ ε * Fintype.card N)
    34 (hc : (8 : ℝ) * Fintype.card Comp ≤ ε * Fintype.card N)
    35 (hσ : ∀ o, σ o ≤ (2 : ℝ≥0∞) ^ (D * Fintype.card N) * PMF.uniformOfFintype _ o) :
    36 ∃ L : Finset (Finset (Fin 2 → Comp → Frame B H N E) × Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N),
    37 Partition σ observation P₀ R ((2 : ℝ≥0∞) ^ (-(ζ * Fintype.card N)))
    38 ((2 : ℝ≥0∞) ^ (-((1 - ζ) * Fintype.card N))) L ∧
    39 ∀ z ∈ L, z.2.rank ≤ P₀.rank + K ∧
    40 LawBound σ (PMF.uniformOfFintype _) observation z.2 z.1
    41 ((2 : ℝ≥0∞) ^ (-((1 - ζ) * Fintype.card N)))
    42 ((2 : ℝ≥0∞) ^ ((D + K + 1) * Fintype.card N))
    43
    44end Lax342547.RawPeeling
    45
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