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Finite accepting-family losses

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

proven

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    Natural Language Statement

    Lemma

    Conditional event identities and finite family unions charge small accepted sets once and retain the actual accepting-pair mass.

    Concept map
    14 concepts; 5 descendants hidden
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    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 accepting_family_loss proven

    2 conditional_event_mass proven

    3 conditional_probability proven

    Lean source view on GitHub

    1import Lax342547.FiniteSampling
    2import Lax342547.RetainedImages
    3
    4/-!
    5---
    6title: Finite accepting-family losses
    7type: lemma
    8---
    9Conditional event identities and finite family unions charge small accepted sets once and retain the actual accepting-pair mass.
    10-/
    11
    12namespace Lax342547.AcceptingFamilies
    13
    14open Lax342547.RelativeEntropy Lax342547.RetainedImages Lax342547.FiniteSampling
    15open scoped BigOperators
    16
    17axiom conditional_event_mass {A : Type} [Fintype A] (α : A → ℝ) (C F : A → Prop) :
    18 cellMass (conditionalLaw α C) F = cellMass α (fun a => C a ∧ F a) / cellMass α C
    19
    20axiom conditional_probability {A : Type} [Fintype A] (α : A → ℝ) (C : A → Prop)
    21 (hα : Probability α) (hC : 0 < cellMass α C) : Probability (conditionalLaw α C)
    22
    23axiom accepting_family_loss {A B I : Type} [Fintype A] [Fintype B] [Fintype I]
    24 (α : A → ℝ) (β : B → ℝ) (C : I → A → Prop) (key : B → I) (F : B → A → Prop)
    25 (η τ U : ℝ) (hα : Probability α) (hβ : Probability β)
    26 (hη : 0 ≤ η) (hτ : 0 < τ) (hU : 0 ≤ U)
    27 (hex : ∀ i, τ ≤ cellMass α (C i) →
    28 cellMass β (fun b => η ≤ cellMass (conditionalLaw α (C i)) (F b)) ≤ U) :
    29 (∑ b, β b * cellMass α (fun a => C (key b) a ∧ F b a)) ≤
    30 η * (∑ b, β b * cellMass α (C (key b))) + τ + Fintype.card I * U
    31
    32end Lax342547.AcceptingFamilies
    33
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