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Fresh peeling after restrictions and deliberate extra pins

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

proven

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    Natural Language Statement

    Lemma

    Every retained metadata fiber carries its old pins and all new prescribed images. Its absolute density pays both preceding normalizations. A fresh peeling fits inside the paper's final total pin budget.

    Concept map
    11 concepts; 1 descendant hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 final_rank_budget proven

    2 second_part proven

    Lean source view on GitHub

    1import Lax342547.SplitTrimming
    2import Lax342547.BoundedPeeling
    3import Mathlib.Algebra.Order.Floor.Ring
    4
    5/-!
    6---
    7title: Fresh peeling after restrictions and deliberate extra pins
    8type: lemma
    9---
    10Every retained metadata fiber carries its old pins and all new prescribed
    11images. Its absolute density pays both preceding normalizations. A fresh
    12peeling fits inside the paper's final total pin budget.
    13-/
    14
    15namespace Lax342547.SecondPeeling
    16
    17open Lax342547.MomentSpace Lax342547.ExactPins Lax342547.MetadataPins
    18open Lax342547.LeafPartition Lax342547.BoundedPeeling
    19open scoped ENNReal
    20
    21def startCost (D : ℝ) (K₁ u : ℕ) : ℝ := D + K₁ + u + 5
    22
    23noncomputable def newBudget (D ζ : ℝ) (K₁ u : ℕ) : ℕ :=
    24 ⌈4 * (startCost D K₁ u + 1) / ζ⌉₊
    25
    26noncomputable def finalBudget (D ζ : ℝ) (K₁ u : ℕ) : ℕ :=
    27 ⌈10 * (D + K₁ + u + 10) / ζ⌉₊
    28
    29noncomputable def supportFinset {Ω : Type} [Fintype Ω] (p : PMF Ω) : Finset Ω := by
    30 classical
    31 exact Finset.univ.filter (fun o => o ∈ p.support)
    32
    33def RepeeledPart {Ω J Axis I N : Type} [Fintype Ω] [DecidableEq Ω] [Fintype Axis] [Fintype N]
    34 (q μ : PMF Ω) (A : Ω → Axis → (I → Binary) →ₗ[Binary] (N → Binary))
    35 (P₀ : Pin Axis I N) (choice : Ω → J) (U : J → Axis → Submodule Binary (I → Binary))
    36 (D ζ : ℝ) (K₁ u : ℕ) (k : Key N U) : Prop :=
    37 ∃ hkSupport : ∃ o ∈ {o | record A choice U o = k}, o ∈ q.support,
    38 let Q := q.filter {o | record A choice U o = k} hkSupport
    39 ∃ P : Pin Axis I N, ∃ L : Finset (Finset Ω × Pin Axis I N),
    40 P₀.Extends P ∧ (pin k).Extends P ∧ P.rank ≤ K₁ + u ∧
    41 Partition Q A P (supportFinset Q)
    42 ((2 : ℝ≥0∞) ^ (-(ζ * Fintype.card N)))
    43 ((2 : ℝ≥0∞) ^ (-((1 - ζ) * Fintype.card N))) L ∧
    44 q.toOuterMeasure (supportFinset Q \ L.biUnion Prod.fst) ≤
    45 (2 : ℝ≥0∞) ^ (-(ζ * Fintype.card N)) *
    46 q.toOuterMeasure {o | record A choice U o = k} ∧
    47 ∀ z ∈ L, z.2.rank ≤ finalBudget D ζ K₁ u ∧
    48 LawBound Q μ A z.2 z.1
    49 ((2 : ℝ≥0∞) ^ (-((1 - ζ) * Fintype.card N)))
    50 ((2 : ℝ≥0∞) ^ ((startCost D K₁ u + newBudget D ζ K₁ u + 1) * Fintype.card N))
    51
    52axiom final_rank_budget (D ζ : ℝ) (K₁ u : ℕ)
    53 (hD : 0 ≤ D) (hζ : 0 < ζ) (hζ₁ : ζ ≤ 1) :
    54 K₁ + u + newBudget D ζ K₁ u ≤ finalBudget D ζ K₁ u
    55
    56axiom second_part {Ω J Axis I N : Type}
    57 [Fintype Ω] [DecidableEq Ω] [Fintype Axis] [Fintype I] [Fintype N]
    58 (p μ : PMF Ω) (A : Ω → Axis → (I → Binary) →ₗ[Binary] (N → Binary))
    59 (P₀ : Pin Axis I N) (S : Set Ω) (hS : ∃ o ∈ S, o ∈ p.support)
    60 (choice : Ω → J) (U : J → Axis → Submodule Binary (I → Binary))
    61 (D ζ ε : ℝ) (K₁ u : ℕ)
    62 (hD : 0 ≤ D) (hζ : 0 < ζ) (hζ₁ : ζ ≤ 1) (hε : ε ≤ ζ / 4)
    63 (hN : 0 < Fintype.card N) (hP : p.support ⊆ P₀.event A) (hPbudget : P₀.rank ≤ K₁)
    64 (hu : ∀ j, ∑ a, Module.finrank Binary (U j a) ≤ u)
    65 (hSbound : (2 : ℝ≥0∞) ^ (-(ζ * Fintype.card N)) ≤ p.toOuterMeasure S)
    66 (hp : ∀ o, p o ≤ (2 : ℝ≥0∞) ^ ((D + K₁ + 1) * Fintype.card N) * μ o)
    67 (hreference : ∀ P : Pin Axis I N, μ.toOuterMeasure (P.event A) ≤
    68 (2 : ℝ≥0∞) ^ (-((1 - ε) * P.rank * Fintype.card N)))
    69 (k : Key N U)
    70 (hk : (2 : ℝ≥0∞) ^ (-(((u : ℝ) + 1) * Fintype.card N)) ≤
    71 (p.filter S hS).toOuterMeasure {o | record A choice U o = k}) :
    72 RepeeledPart (p.filter S hS) μ A P₀ choice U D ζ K₁ u k
    73
    74end Lax342547.SecondPeeling
    75
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