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Deterministic entropy and neighborhood selections

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

proven

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    Natural Language Statement

    Lemma

    Compact feasible residual laws have fixed entropy minimizers. Raw supersaturation supplies a deterministic residual unit with positive neighborhood mass, guaranteeing progress of fingerprint deletion.

    Concept map
    13 concepts; 10 descendants 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 entropy_selectors proven

    2 heavy_neighbor_nonempty proven

    Lean source view on GitHub

    1import Lax342547.ContainerLaws
    2
    3/-!
    4---
    5title: Deterministic entropy and neighborhood selections
    6type: lemma
    7---
    8Compact feasible residual laws have fixed entropy minimizers. Raw supersaturation supplies a deterministic residual unit with positive neighborhood mass, guaranteeing progress of fingerprint deletion.
    9-/
    10
    11namespace Lax342547.EntropySelectors
    12
    13open Lax342547.RelativeEntropy Lax342547.ContainerLaws Lax342547.RetainedImages
    14open scoped BigOperators
    15
    16axiom heavy_neighbor_nonempty {U : Type} [Fintype U]
    17 (ρ : U → ℝ) (R : Finset U) (H : U → U → Prop) (v : U)
    18 (hs : ∀ x, x ∉ R → ρ x = 0) (hm : 0 < neighborMass ρ H v) : ∃ w ∈ R, H v w
    19
    20axiom entropy_selectors {U : Type} [Fintype U] [Nonempty U]
    21 (P : Set (U → ℝ)) (q : U → ℝ) (H : U → U → Prop) (ε : ℝ)
    22 (hc : IsCompact P) (hprob : ∀ ρ ∈ P, Probability ρ) (hε : 0 < ε)
    23 (hsuper : ∀ ρ ∈ P, ε ≤ conflictMass ρ H) :
    24 ∃ law : Finset U → U → ℝ, ∃ pick : Finset U → U,
    25 ∀ R, (residual P R).Nonempty →
    26 law R ∈ residual P R ∧
    27 (∀ σ ∈ residual P R, entropy (law R) q ≤ entropy σ q) ∧
    28 pick R ∈ R ∧ ε ≤ neighborMass (law R) H (pick R) ∧
    29 (∃ w ∈ R, H (pick R) w)
    30
    31end Lax342547.EntropySelectors
    32
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