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Entropy bounded container families

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

proven

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    Natural Language Statement

    Lemma

    Compact convex feasible laws with an entropy budget and uniform conflict supersaturation admit a finite family of infeasible containers covering every independent set. Ordered fingerprints obey the same exponential counting budget as the paper.

    Concept map
    16 concepts; 9 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 entropy_container_family proven

    2 entropy_fingerprint_length proven

    3 entropy_increment_positive proven

    Lean source view on GitHub

    1import Lax342547.EntropySelectors
    2import Lax342547.ContainerRun
    3import Lax342547.FingerprintCounts
    4
    5/-!
    6---
    7title: Entropy bounded container families
    8type: lemma
    9---
    10Compact convex feasible laws with an entropy budget and uniform conflict supersaturation admit a finite family of infeasible containers covering every independent set. Ordered fingerprints obey the same exponential counting budget as the paper.
    11-/
    12
    13namespace Lax342547.EntropyContainers
    14
    15open Lax342547.RelativeEntropy Lax342547.ContainerLaws Lax342547.ContainerRun
    16open scoped BigOperators
    17
    18axiom entropy_increment_positive (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1) : 0 < -Real.log (1-ε)
    19
    20axiom entropy_fingerprint_length {U : Type} [Fintype U]
    21 (P : Set (U → ℝ)) (q : U → ℝ) (H : U → U → Prop) (ε B : ℝ)
    22 (law : Finset U → U → ℝ) (pick : Finset U → U) (I R : Finset U) (fuel L : ℕ)
    23 (hprob : ∀ ρ ∈ P, Probability ρ) (hc : Convex ℝ P)
    24 (hb : ∀ ρ ∈ P, 0 ≤ entropy ρ q ∧ entropy ρ q ≤ B)
    25 (hε : 0 < ε) (hε1 : ε < 1)
    26 (hspec : ∀ T, (residual P T).Nonempty →
    27 law T ∈ residual P T ∧ (∀ σ ∈ residual P T, entropy (law T) q ≤ entropy σ q) ∧
    28 pick T ∈ T ∧ ε ≤ neighborMass (law T) H (pick T) ∧ (∃ w ∈ T, H (pick T) w))
    29 (hL : B/(-Real.log (1-ε))+1 ≤ L) (hactive : (residual P R).Nonempty) :
    30 (run (fun T => (residual P T).Nonempty) pick H I fuel R).2.length ≤ L
    31
    32axiom entropy_container_family {U : Type} [Fintype U] [Nonempty U]
    33 (P : Set (U → ℝ)) (q : U → ℝ) (H : U → U → Prop) (ε B : ℝ) (L : ℕ)
    34 (hcompact : IsCompact P) (hconvex : Convex ℝ P) (hprob : ∀ ρ ∈ P, Probability ρ)
    35 (hbudget : ∀ ρ ∈ P, 0 ≤ entropy ρ q ∧ entropy ρ q ≤ B)
    36 (hε : 0 < ε) (hε1 : ε < 1)
    37 (hL : B/(-Real.log (1-ε))+1 ≤ L)
    38 (hsuper : ∀ ρ ∈ P, ε ≤ conflictMass ρ H) :
    39 ∃ C : Finset (Finset U),
    40 C.card ≤ (Fintype.card U+1)^L ∧
    41 (∀ R ∈ C, ¬ (residual P R).Nonempty) ∧
    42 (∀ I : Finset U, (∀ x ∈ I, ∀ y ∈ I, ¬ H x y) → ∃ R ∈ C, I ⊆ R)
    43
    44end Lax342547.EntropyContainers
    45
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