CorollaryStatement: AI-generatedProof: AI-generatedPipeline-generatedprecheck passjudge 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.

The compact-space WGAN value is differentiable almost everywhere under choice

Statement

Assume the full Axiom of Choice and all hypotheses of the common-neighborhood integrable-envelope branch of the compact-space WGAN theorem. Then its value function $V : Θ \to R$ is Fréchet differentiable outside a Lebesgue-null subset of $Θ$ .

Statement references

Axiom of Choice

Compact-space WGAN continuity and local-Lipschitz theorem

Euclidean Rademacher differentiability theorem

Facts & Assumptions

Given: The full Axiom of Choice, an open $Θ \subseteq R^{p}$ with $p \geq 1$ , and the hypotheses of the common-neighborhood integrable-envelope branch of the WGAN theorem.

[A1]

The Axiom of Choice states that every family of nonempty sets has a choice function. The Axiom of Choice

[F1]

Under its common-neighborhood integrable-envelope hypothesis, the WGAN value $V$ is locally Lipschitz on $Θ$ . Continuity and ideal-critic gradients for compact-space Wasserstein GAN objectives

[F2]

Assuming full AC, every locally Lipschitz map from an open subset of $R^{p}$ to $R$ is Fréchet differentiable outside a Lebesgue-null subset. Rademacher differentiability in Euclidean space

Proof

1.1givenF1

By [F1], $V$ is a locally Lipschitz real-valued function on the open set $Θ \subseteq R^{p}$ .

2.1givenA1F2step 1.1∎

Under [A1], apply [F2] to $V$ on $Θ$ ; it is Fréchet differentiable outside a Lebesgue-null subset of $Θ$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

39 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

Martin Arjovsky, Soumith Chintala and Léon Bottou, Wasserstein GAN, Theorem 1 and Appendix C, physical PDF pp. 5 and 20–21 (standard reference, not scraped)