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Two-batch comparison across independent component groups

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

proven

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    Natural Language Statement

    Lemma

    Replacing one independent group at a time adds its bounded separate-test error. The global tests may depend arbitrarily on all groups, and the two observations inside each group retain their full joint dependence. Different groups may have different finite state spaces and column counts.

    Concept map
    23 concepts; 8 descendants hidden
    100%
    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 product_comparison proven

    2 reference_product_comparison proven

    3 uniform_group_comparison proven

    Lean source view on GitHub

    1import Lax342547.RawFrameComparison
    2
    3/-!
    4---
    5title: Two-batch comparison across independent component groups
    6type: lemma
    7---
    8Replacing one independent group at a time adds its bounded separate-test
    9error. The global tests may depend arbitrarily on all groups, and the two
    10observations inside each group retain their full joint dependence. Different
    11groups may have different finite state spaces and column counts.
    12-/
    13
    14namespace Lax342547.GroupTensorComparison
    15noncomputable section
    16open Lax342547.RelativeEntropy
    17open scoped BigOperators
    18variable {ι : Type} [Fintype ι] [DecidableEq ι]
    19 {X Y : ι → Type} [∀ i,Fintype (X i)] [∀ i,Fintype (Y i)]
    20
    21def mean (ρ : ∀ i,X i × Y i → ℝ) (f : (∀ i,X i) → ℝ) (g : (∀ i,Y i) → ℝ) : ℝ :=
    22 ∑ z : ∀ i,X i × Y i,(∏ i,ρ i (z i))*f (fun i => (z i).1)*g (fun i => (z i).2)
    23
    24axiom product_comparison (ρ σ : ∀ j,X j × Y j → ℝ)
    25 (hρ : ∀ j,Probability (ρ j)) (hσ : ∀ j,Probability (σ j)) (ε : ι → ℝ)
    26 (hc : ∀ i,∀ f : X i → ℝ,∀ g : Y i → ℝ,(∀ x,|f x| ≤ 1) → (∀ y,|g y| ≤ 1) →
    27 |(∑ z : X i × Y i,ρ i z*f z.1*g z.2)-(∑ z : X i × Y i,σ i z*f z.1*g z.2)| ≤ ε i)
    28 (f : (∀ j,X j) → ℝ) (g : (∀ j,Y j) → ℝ)
    29 (hf : ∀ x,|f x| ≤ 1) (hg : ∀ y,|g y| ≤ 1) :
    30 |mean ρ f g-mean σ f g| ≤ ∑ i,ε i
    31
    32axiom reference_product_comparison (ρ : ∀ i,X i × Y i → ℝ)
    33 (α : ∀ i,X i → ℝ) (β : ∀ i,Y i → ℝ)
    34 (hρ : ∀ i,Probability (ρ i)) (hα : ∀ i,Probability (α i)) (hβ : ∀ i,Probability (β i))
    35 (ε : ι → ℝ)
    36 (hc : ∀ i,∀ f : X i → ℝ,∀ g : Y i → ℝ,(∀ x,|f x| ≤ 1) → (∀ y,|g y| ≤ 1) →
    37 |(∑ z : X i × Y i,ρ i z*f z.1*g z.2)-
    38 (∑ x,α i x*f x)*(∑ y,β i y*g y)| ≤ ε i)
    39 (f : (∀ i,X i) → ℝ) (g : (∀ i,Y i) → ℝ)
    40 (hf : ∀ x,|f x| ≤ 1) (hg : ∀ y,|g y| ≤ 1) :
    41 |mean ρ f g-(∑ x,(∏ i,α i (x i))*f x)*(∑ y,(∏ i,β i (y i))*g y)| ≤ ∑ i,ε i
    42
    43end
    44noncomputable section
    45variable {ι : Type} [Fintype ι] [DecidableEq ι]
    46 {X Y Ω : ι → Type} [∀ i,Fintype (Ω i)] [∀ i,Nonempty (Ω i)]
    47open Lax342547.RealCellLaws
    48
    49axiom uniform_group_comparison [∀ i,Fintype (X i)] [∀ i,Fintype (Y i)] (x : ∀ i,Ω i → X i) (y : ∀ i,Ω i → Y i)
    50 (ε : ι → ℝ)
    51 (hc : ∀ i,∀ f : X i → ℝ,∀ g : Y i → ℝ,(∀ x,|f x| ≤ 1) → (∀ y,|g y| ≤ 1) →
    52 |(∑ o : Ω i,weights (PMF.uniformOfFintype _) o*(f (x i o)*g (y i o)))-
    53 (∑ o : Ω i,weights (PMF.uniformOfFintype _) o*f (x i o))*
    54 (∑ o : Ω i,weights (PMF.uniformOfFintype _) o*g (y i o))| ≤ ε i)
    55 (f : (∀ i,X i) → ℝ) (g : (∀ i,Y i) → ℝ)
    56 (hf : ∀ x,|f x| ≤ 1) (hg : ∀ y,|g y| ≤ 1) :
    57 |(∑ o : ∀ i,Ω i,weights (PMF.uniformOfFintype _) o*
    58 (f (fun i => x i (o i))*g (fun i => y i (o i))))-
    59 (∑ o : ∀ i,Ω i,weights (PMF.uniformOfFintype _) o*f (fun i => x i (o i)))*
    60 (∑ o : ∀ i,Ω i,weights (PMF.uniformOfFintype _) o*g (fun i => y i (o i)))| ≤ ∑ i,ε i
    61
    62end
    63end Lax342547.GroupTensorComparison
    64
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