Lax Lean Archive

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

Second exact-pin peeling for the actual raw frame law

Lax342547.RawSecondPeeling · concepts/Lax342547/RawSecondPeeling.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

    Theorem

    After a retained restriction of an old leaf, split by bounded coefficient metadata and exact images. Discard exponentially small mass and apply a fresh peeling on every retained fiber, within the paper's final budget.

    Concept map
    26 concepts
    100%
    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.SecondPeeling
    2import Lax342547.ReferencePins
    3
    4/-!
    5---
    6title: Second exact-pin peeling for the actual raw frame law
    7type: theorem
    8---
    9After a retained restriction of an old leaf, split by bounded coefficient
    10metadata and exact images. Discard exponentially small mass and apply a
    11fresh peeling on every retained fiber, within the paper's final budget.
    12-/
    13
    14namespace Lax342547.RawSecondPeeling
    15
    16open Lax342547.MomentSpace Lax342547.RawFrames Lax342547.ExactPins
    17open Lax342547.ReferencePins Lax342547.MetadataPins Lax342547.SecondPeeling
    18open scoped ENNReal
    19
    20axiom raw_second_peeling {J Comp B H N : Type}
    21 [Fintype J] [Fintype Comp] [Fintype B] [Fintype H] [Fintype N]
    22 [DecidableEq Comp] [DecidableEq B] [DecidableEq H] [DecidableEq N]
    23 {E : Matrix B B Binary} [Nonempty (Frame B H N E)] [DecidableEq (Frame B H N E)]
    24 (p : PMF (Fin 2 → Comp → Frame B H N E))
    25 (P₀ : Pin (Comp × Bool) (Fin 2 × (B ⊕ H)) N)
    26 (S : Set (Fin 2 → Comp → Frame B H N E)) (hS : ∃ o ∈ S, o ∈ p.support)
    27 (choice : (Fin 2 → Comp → Frame B H N E) → J)
    28 (U : J → (Comp × Bool) → Submodule Binary ((Fin 2 × (B ⊕ H)) → Binary))
    29 (D ζ ε : ℝ) (K₁ u a n : ℕ)
    30 (hD : 0 ≤ D) (hζ : 0 < ζ) (hζ₁ : ζ ≤ 1) (hε : 0 < ε) (hεζ : ε ≤ ζ / 4)
    31 (hN : 2 * (Fintype.card B + Fintype.card H) + 1 ≤ Fintype.card N)
    32 (hcoeff : (4 : ℝ) * (Fintype.card B + Fintype.card H : ℕ) ≤ ε * Fintype.card N)
    33 (hcomp : (8 : ℝ) * Fintype.card Comp ≤ ε * Fintype.card N)
    34 (hP : p.support ⊆ P₀.event observation) (hPbudget : P₀.rank ≤ K₁)
    35 (hu : ∀ j, ∑ b, Module.finrank Binary (U j b) ≤ u)
    36 (hchoices : Fintype.card J ≤ 2 ^ (a * n)) (hscale : 2 * a * n ≤ Fintype.card N)
    37 (hSbound : (2 : ℝ≥0∞) ^ (-(ζ * Fintype.card N)) ≤ p.toOuterMeasure S)
    38 (hp : ∀ o, p o ≤ (2 : ℝ≥0∞) ^ ((D + K₁ + 1) * Fintype.card N) * PMF.uniformOfFintype _ o) :
    39 (p.filter S hS).toOuterMeasure {o |
    40 (p.filter S hS).toOuterMeasure {x | record observation choice U x = record observation choice U o} <
    41 (2 : ℝ≥0∞) ^ (-(((u : ℝ) + 1) * Fintype.card N))} ≤
    42 (2 : ℝ≥0∞) ^ (-(Fintype.card N : ℝ) / 2) ∧
    43 ∀ k : Key N U,
    44 (2 : ℝ≥0∞) ^ (-(((u : ℝ) + 1) * Fintype.card N)) ≤
    45 (p.filter S hS).toOuterMeasure {o | record observation choice U o = k} →
    46 RepeeledPart (p.filter S hS) (PMF.uniformOfFintype _) observation P₀ choice U D ζ K₁ u k
    47
    48end Lax342547.RawSecondPeeling
    49
    Show Proof
    Builds on
    Used by

    none

    From Mathlib

    none

    Discussion

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

    Loading discussion…