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Admissible pair laws

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

proven

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    Natural Language Statement

    Lemma

    Compactness, convexity and entropy minimization for the actual marginal and pair capacity constraints.

    Concept map
    11 concepts
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    This concept declares 5 statements. Each proof establishes one of them relative to its assumptions.

    1 entropy_minimizer proven

    2 feasible_closed proven

    3 feasible_compact proven

    4 feasible_convex proven

    5 feasible_entropy_budget proven

    Lean source view on GitHub

    1import Lax342547.EntropyProjection
    2import Lax342547.PairRecovery
    3
    4/-!
    5---
    6title: Admissible pair laws
    7type: lemma
    8---
    9Compactness, convexity and entropy minimization for the actual marginal and pair capacity constraints.
    10-/
    11
    12namespace Lax342547.UnitLaws
    13
    14open Lax342547.RelativeEntropy
    15open scoped BigOperators
    16
    17def Feasible {Ω : Type} [Fintype Ω] (μ : Ω → ℝ) (R : Ω → Ω → Prop)
    18 (M κ : ℝ) (ρ : Ω × Ω → ℝ) : Prop :=
    19 Probability ρ ∧ (∀ x, (∑ y, ρ (x,y)) ≤ M*μ x) ∧
    20 (∀ y, (∑ x, ρ (x,y)) ≤ M*μ y) ∧
    21 (∀ xy, ρ xy ≤ κ*(μ xy.1*μ xy.2)) ∧ (∀ xy, ¬ R xy.1 xy.2 → ρ xy = 0)
    22
    23axiom feasible_closed {Ω : Type} [Fintype Ω]
    24 (μ : Ω → ℝ) (R : Ω → Ω → Prop) (M κ : ℝ) : IsClosed {ρ | Feasible μ R M κ ρ}
    25
    26axiom feasible_compact {Ω : Type} [Fintype Ω]
    27 (μ : Ω → ℝ) (R : Ω → Ω → Prop) (M κ : ℝ) : IsCompact {ρ | Feasible μ R M κ ρ}
    28
    29axiom feasible_convex {Ω : Type} [Fintype Ω]
    30 (μ : Ω → ℝ) (R : Ω → Ω → Prop) (M κ : ℝ) : Convex ℝ {ρ | Feasible μ R M κ ρ}
    31
    32axiom entropy_minimizer {Ω : Type} [Fintype Ω]
    33 (μ : Ω → ℝ) (R : Ω → Ω → Prop) (M κ : ℝ)
    34 (hne : ∃ ρ, Feasible μ R M κ ρ) :
    35 ∃ ρ, Feasible μ R M κ ρ ∧
    36 (∀ ρ', Feasible μ R M κ ρ' → entropy ρ (fun xy => μ xy.1*μ xy.2) ≤ entropy ρ' (fun xy => μ xy.1*μ xy.2)) ∧
    37 (∀ ρ', Feasible μ R M κ ρ' → ∀ xy, 0 < ρ' xy → 0 < ρ xy) ∧
    38 (∀ ρ', Feasible μ R M κ ρ' → entropy ρ' ρ ≤
    39 entropy ρ' (fun xy => μ xy.1*μ xy.2)-entropy ρ (fun xy => μ xy.1*μ xy.2))
    40
    41axiom feasible_entropy_budget {Ω : Type} [Fintype Ω]
    42 (μ : Ω → ℝ) (R : Ω → Ω → Prop) (M κ : ℝ) (ρ : Ω × Ω → ℝ)
    43 (hμ : Probability μ) (hμpos : ∀ x, 0 < μ x) (hκ : 0 < κ) (hρ : Feasible μ R M κ ρ) :
    44 0 ≤ entropy ρ (fun xy => μ xy.1*μ xy.2) ∧ entropy ρ (fun xy => μ xy.1*μ xy.2) ≤ Real.log κ
    45
    46end Lax342547.UnitLaws
    47
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