Provedsampleability-lawtile-proposalgpuchi-square

Tile marginalization data processing for the chi-square cost factor

For all $\htmlData{mx=1}{J, T \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{\mathit{tile} : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{T}}$ and $\htmlData{mx=3}{\pi, r : \htmlData{mx=0}{J} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ , if $\htmlData{mx=4}{\forall j,\ 0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{r}(\htmlData{mx=1}{j})}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{J}}$ , $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{T}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{DecidableEq}}\,\htmlData{mx=1}{T}}$ ), then

\htmlData{mx=26}{\sum_{t} \htmlData{mx=25}{\frac{\htmlData{mx=12}{\left(\htmlData{mx=11}{\sum_{j \in \htmlData{mx=7}{\htmlData{mx=0}{\operatorname{filter}}(\htmlData{mx=6}{j \mapsto \htmlData{mx=3}{\htmlData{mx=1}{\mathit{tile}}(\htmlData{mx=2}{j})} \mathrel{\htmlData{mx=5}{=}} \htmlData{mx=4}{t}})}} \htmlData{mx=10}{\htmlData{mx=8}{\pi}(\htmlData{mx=9}{j})}}\right)^{2}}}{\htmlData{mx=24}{\sum_{j \in \htmlData{mx=20}{\htmlData{mx=13}{\operatorname{filter}}(\htmlData{mx=19}{j \mapsto \htmlData{mx=16}{\htmlData{mx=14}{\mathit{tile}}(\htmlData{mx=15}{j})} \mathrel{\htmlData{mx=18}{=}} \htmlData{mx=17}{t}})}} \htmlData{mx=23}{\htmlData{mx=21}{r}(\htmlData{mx=22}{j})}}}}} \mathrel{\htmlData{mx=36}{\leq}} \htmlData{mx=35}{\sum_{j} \htmlData{mx=34}{\frac{\htmlData{mx=30}{\htmlData{mx=29}{\htmlData{mx=27}{\pi}(\htmlData{mx=28}{j})}^{2}}}{\htmlData{mx=33}{\htmlData{mx=31}{r}(\htmlData{mx=32}{j})}}}}

What the notation means · or point at any symbol

Type: Type: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Type)
x ∈ TVariables: J, T range over Type: the statement must hold for every choice. ((J T : Type))
Fintype: Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application: Fintype applied to J. Lean writes f x; papers write f(x). (Fintype J)
DecidableEq: DecidableEq: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (DecidableEq)
⟹Implies: If J, then T. (Between types, A → B is the type of functions from A to B.) (J → T)
x : TVariables: π, r range over J → ℝ: the statement must hold for every choice. ((π r : J → ℝ))
∀For all: The statement after the comma holds for every j. (∀ j, 0 < r j)
≤At most: ∑ t, (∑ j, π(j))^2/(∑ j, r(j)) is less than or equal to ∑ j, π(j)^2/r(j). (∑ t, (∑ j ∈ Finset.univ.filter (fun j =…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
<Less than: 0 is strictly smaller than r(j). (0 < r j)
aᵇPower: ∑ j, π(j) raised to the power 2. ((∑ j ∈ Finset.univ.filter (fun j => til…)
a/bDivision: (∑ j, π(j))^2 divided by ∑ j, r(j). In Lean, dividing by 0 gives 0. ((∑ j ∈ Finset.univ.filter (fun j => til…)
∑Summation: Add up π(j) for each j in filter(j ↦ tile(j) = t). (∑ j ∈ Finset.univ.filter (fun j => tile…)
filter: Finset.univ.filter: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.univ.filter)
x ↦ f(x)Function (λ-abstraction): The function sending j to tile(j) = t. Lean writes fun x => …; papers write x ↦ …. (fun j => tile j = t)
=Equality: tile(j) and t are the same. (tile j = t)
πPi: The ratio of a circle's circumference to its diameter, ≈ 3.14159. (π)

theorem tile-proposal-data-processing {J T : Type} [Fintype J] [Fintype T] [DecidableEq T] (tile : J → T) (π r : J → ℝ) (hr : ∀ j, 0 < r j) :
    ∑ t, (∑ j ∈ Finset.univ.filter (fun j => tile j = t), π j) ^ 2 / (∑ j ∈ Finset.univ.filter (fun j => tile j = t), r j) ≤ ∑ j, π j ^ 2 / r j

Verified inequality concerns marginal target and proposal masses over a partition. It does not price within-tile computation or key sampling, and does not resolve GPU access efficiency. Uniform within-tile selection changes the proposal and can increase the full factor. Exact tile aggregation requires all tile scores and value reads to be charged.

How it was proved

Proved: 5 October 2026
By: Ajax, with the agent “Baguette math & proofs (Claude Code, 2026-10-03)”
Checked by: Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms: Classical.choice, Quot.sound, propext
Proof: 12 lines of Lean

Tile marginalization data processing for the chi-square cost factor

How it was proved

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