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Entropy progress for residual pair laws

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

proven

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    Natural Language Statement

    Lemma

    Residual feasible families retain compactness and convexity. Supersaturation selects a heavy neighborhood, and deleting it increases the minimum entropy by the exact negative log support cost.

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

    This concept declares 10 statements. Each proof establishes one of them relative to its assumptions.

    1 heavy_neighbor proven

    2 mass_complement proven

    3 minimum_mono proven

    4 positive_support proven

    5 recording_entropy_increment proven

    6 residual_compact proven

    7 residual_convex proven

    8 residual_mono proven

    9 support_mass_positive proven

    10 supported_mass proven

    Lean source view on GitHub

    1import Lax342547.UnitLaws
    2
    3/-!
    4---
    5title: Entropy progress for residual pair laws
    6type: lemma
    7---
    8Residual feasible families retain compactness and convexity. Supersaturation selects a heavy neighborhood, and deleting it increases the minimum entropy by the exact negative log support cost.
    9-/
    10
    11namespace Lax342547.ContainerLaws
    12
    13open Lax342547.RelativeEntropy Lax342547.RetainedImages
    14open scoped BigOperators
    15
    16noncomputable def residual {U : Type} [Fintype U] (P : Set (U → ℝ)) (R : Finset U) : Set (U → ℝ) :=
    17 {ρ | ρ ∈ P ∧ ∀ x, x ∉ R → ρ x = 0}
    18
    19noncomputable def neighborMass {U : Type} [Fintype U] (ρ : U → ℝ) (H : U → U → Prop) (v : U) : ℝ :=
    20 cellMass ρ (H v)
    21
    22noncomputable def conflictMass {U : Type} [Fintype U] (ρ : U → ℝ) (H : U → U → Prop) : ℝ :=
    23 ∑ v, ρ v * neighborMass ρ H v
    24
    25axiom residual_mono {U : Type} [Fintype U] (P : Set (U → ℝ)) (R T : Finset U)
    26 (hTR : T ⊆ R) : residual P T ⊆ residual P R
    27
    28axiom residual_compact {U : Type} [Fintype U] (P : Set (U → ℝ)) (R : Finset U)
    29 (hP : IsCompact P) : IsCompact (residual P R)
    30
    31axiom residual_convex {U : Type} [Fintype U] (P : Set (U → ℝ)) (R : Finset U)
    32 (hP : Convex ℝ P) : Convex ℝ (residual P R)
    33
    34axiom supported_mass {U : Type} [Fintype U] (ρ : U → ℝ) (R : Finset U)
    35 (hρ : Probability ρ) (hs : ∀ x, x ∉ R → ρ x = 0) : cellMass ρ (fun x => x ∈ R) = 1
    36
    37axiom positive_support {U : Type} [Fintype U] (ρ : U → ℝ) (hρ : Probability ρ) :
    38 ∃ x, 0 < ρ x
    39
    40axiom mass_complement {U : Type} [Fintype U] (ρ : U → ℝ) (S : U → Prop)
    41 (hρ : Probability ρ) : cellMass ρ (fun x => ¬ S x) = 1-cellMass ρ S
    42
    43axiom support_mass_positive {U : Type} [Fintype U]
    44 (ρ σ : U → ℝ) (S : U → Prop) (hρ : Probability ρ) (hσ : Probability σ)
    45 (hs : ∀ x, ¬ S x → σ x = 0) (hfull : ∀ x, 0 < σ x → 0 < ρ x) :
    46 0 < cellMass ρ S
    47
    48axiom heavy_neighbor {U : Type} [Fintype U] (ρ : U → ℝ) (H : U → U → Prop)
    49 (R : Finset U) (ε : ℝ) (hρ : Probability ρ) (hs : ∀ x, x ∉ R → ρ x = 0)
    50 (_hε : 0 < ε) (hmass : ε ≤ conflictMass ρ H) :
    51 ∃ v ∈ R, ε ≤ neighborMass ρ H v
    52
    53axiom minimum_mono {U : Type} [Fintype U] (P : Set (U → ℝ)) (q ρ σ : U → ℝ)
    54 (R T : Finset U) (hTR : T ⊆ R) (hσ : σ ∈ residual P T)
    55 (hmin : ∀ τ ∈ residual P R, entropy ρ q ≤ entropy τ q) : entropy ρ q ≤ entropy σ q
    56
    57axiom recording_entropy_increment {U : Type} [Fintype U]
    58 (P : Set (U → ℝ)) (q ρ σ : U → ℝ) (R : Finset U) (H : U → U → Prop) (v : U) (ε : ℝ)
    59 (hprob : ∀ τ ∈ P, Probability τ) (hconv : Convex ℝ P)
    60 (hρ : ρ ∈ residual P R) (hmin : ∀ τ ∈ residual P R, entropy ρ q ≤ entropy τ q)
    61 (hσ : σ ∈ residual P R ∧ ∀ x, H v x → σ x = 0)
    62 (_hε : ε < 1) (hheavy : ε ≤ neighborMass ρ H v) :
    63 -Real.log (1-ε) ≤ entropy σ q-entropy ρ q
    64
    65end Lax342547.ContainerLaws
    66
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