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Independence of distinct sample positions

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

proven

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    Natural Language Statement

    Lemma

    An embedding of selected positions extends to a partition of the whole position space. Summing the unused independent coordinates proves the exact product law on the selected positions.

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

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

    1 complete_left proven

    2 embedding_restriction proven

    3 sum_restriction proven

    Lean source view on GitHub

    1import Lax342547.FiniteSampling
    2import Mathlib.Logic.Equiv.Set
    3
    4/-!
    5---
    6title: Independence of distinct sample positions
    7type: lemma
    8---
    9An embedding of selected positions extends to a partition of the whole position space. Summing the unused independent coordinates proves the exact product law on the selected positions.
    10-/
    11
    12namespace Lax342547.CoordinateRestrictions
    13
    14open Lax342547.RelativeEntropy Lax342547.FiniteSampling Lax342547.RetainedImages
    15open scoped BigOperators
    16
    17noncomputable def complete {J ι : Type} (a : J ↪ ι) : J ⊕ {i : ι // i ∉ Set.range a} ≃ ι := by
    18 classical
    19 exact (Equiv.sumCongr (Equiv.ofInjective a a.injective) (Equiv.refl _)).trans
    20 (Equiv.Set.sumCompl (Set.range a))
    21
    22axiom complete_left {J ι : Type} (a : J ↪ ι) (j : J) : complete a (.inl j) = a j
    23
    24axiom sum_restriction {J K Ω : Type} [Fintype J] [Fintype K] [Fintype Ω]
    25 [DecidableEq J] [DecidableEq K]
    26 (μ : Ω → ℝ) (A : (J → Ω) → Prop) (hμ : Probability μ) :
    27 cellMass (productLaw (fun _ : J ⊕ K => μ)) (fun sample => A (fun j => sample (.inl j))) =
    28 cellMass (productLaw (fun _ : J => μ)) A
    29
    30axiom embedding_restriction {J ι Ω : Type} [Fintype J] [Fintype ι] [Fintype Ω]
    31 [DecidableEq J] [DecidableEq ι]
    32 (μ : Ω → ℝ) (a : J ↪ ι) (A : (J → Ω) → Prop) (hμ : Probability μ) :
    33 cellMass (productLaw (fun _ : ι => μ)) (fun sample => A (fun j => sample (a j))) =
    34 cellMass (productLaw (fun _ : J => μ)) A
    35
    36end Lax342547.CoordinateRestrictions
    37
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