TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Coordinate Poincare inequalities bound total Sobol effects

Statement

Assume Countable Choice and Dependent Choice. Let $X = (X_{1}, \dots, X_{d})$ have a finite product probability law, let $f$ be a real class in $L^{2}$ , and choose a real product-measurable representative $F$ of $f$ . Fix $i$ such that its marginal law is $μ_{i} (A) = λ (A \cap [a_{i}, b_{i}]) / L_{i}$ with $L_{i} = b_{i} - a_{i} > 0$ . Assume that for almost every fixed $x_{- i}$ , the section $x_{i} \mapsto F (x_{i}, x_{- i})$ is absolutely continuous and its derivative $\partial_{i} F$ is square-integrable under the product law. If $V = Var (f (X)) > 0$ , then $S_{T_{i}} \leq \frac{L_{i}^{2}}{π^{2}} \frac{E ∣ \partial_{i} F (X) ∣^{2}}{V} .$ More generally, the same conclusion holds with $L_{i}^{2} / π^{2}$ replaced by any constant $C_{i}$ for which each such almost-everywhere coordinate section of $F$ satisfies ${Var}_{X_{i}} (g) \leq C_{i} E_{X_{i}} ∣ g^{'} ∣^{2}$ .

Facts & Assumptions

Given: Countable Choice and Dependent Choice, the product law, a real product-measurable representative $F$ of a real $f \in L^{2}$ , the regularity assumptions in the Statement, and positive output variance.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every sequence of nonempty sets has a choice function.

[A2]

The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain: for a nonempty set $X$ , an entire relation $R$ on $X$ , and $a \in X$ , there is $x : N \to X$ with $x_{0} = a$ and $x_{n} R x_{n + 1}$ for every $n$ .

[F1]

Independent product inputs have an orthogonal Hoeffding decomposition: $f$ is the orthogonal sum of its product ANOVA components, each $f_{u}$ depends only on $X_{u}$ , and $P_{u ∖ {j}} f_{u} = 0$ for every $j \in u$ .

[F2]

First-order and total Sobol indices: $V S_{T_{i}} = \sum_{u ∋ i} V_{u}$ when $V > 0$ .

[F3]

Sharp Poincare bound for the uniform law on an interval: for the uniform law on an interval of length $L_{i}$ , the coordinate variance of an absolutely continuous section is at most $(L_{i}^{2} / π^{2})$ times its derivative's second moment.

[F4]

Fubini's theorem for L^1 functions on a sigma-finite product and Tonelli's theorem for nonnegative measurable functions on a sigma-finite product: the almost-everywhere coordinate inequalities may be integrated over the remaining product coordinates.

[F5]

Product-coordinate averaging operators: $P_{- i}$ averages only coordinate $i$ and lifts the section average to the full product space.

[F6]

Product-coordinate projections commute and contract in L2: $P_{J} P_{K} = P_{J \cap K}$ for every pair of coordinate subsets.

Proof

Given: The hypotheses, axioms [A1]–[A2], and facts [F1]–[F6].

1.1F1F2F5F6

For $i \notin u$ , [F5] and dependence only on $X_{u}$ give $P_{- i} f_{u} = f_{u}$ . For $i \in u$ , dependence gives $P_{u} f_{u} = f_{u}$ , so [F6] yields $P_{- i} f_{u} = P_{- i} P_{u} f_{u} = P_{u ∖ {i}} f_{u} = 0$ by the coordinate-centering property in [F1]. Linearity and reconstruction in [F1] therefore give $f - P_{- i} f = \sum_{u ∋ i} f_{u}$ . Orthogonality [F1] and the index definition [F2] give $∥ f - P_{- i} f ∥_{2}^{2} = \sum_{u ∋ i} V_{u} = V S_{T_{i}}$ .

2.1A1A2F3F4F5step 1.1

For almost every $x_{- i}$ , apply [F3] to the section of the chosen representative $F$ . Its variance is $\int ∣ F - P_{- i} F ∣^{2} d μ_{i}$ , and its derivative moment is $\int ∣ \partial_{i} F ∣^{2} d μ_{i}$ . Integrate the inequality over $x_{- i}$ ; [F4] justifies the iterated integrals and gives $∥ f - P_{- i} f ∥_{2}^{2} \leq (L_{i}^{2} / π^{2}) E ∣ \partial_{i} F (X) ∣^{2}$ .

3.1F2step 1.1step 2.1givenalgebra∎

Substitute the identity from step 1.1 into step 2.1 and divide by $V > 0$ . This proves the displayed bound. The same calculation with any valid coordinate constant $C_{i}$ proves the stated general form.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

40 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Matieyendou Lamboni, Bertrand Iooss, Anne-Laure Popelin, and Fabrice Gamboa, Derivative-based global sensitivity measures: general links with Sobol' indices and numerical tests, §3 (standard reference, not scraped)
I.M. Sobol' and S. Kucherenko, Derivative based global sensitivity measures and their link with global sensitivity indices, §4, Theorem 2 (standard reference, not scraped)