LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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.

BV extension, translation, and coarea

Statement

Assume Countable Choice. For every bounded Lipschitz domain $Ω \subseteq R^{n}$ (a bounded open set with a finite Lipschitz graph cover of its boundary, with the domain on one side of each graph), zero extension is a bounded map $B V (Ω) \to B V (R^{n})$ with support in $\overline{Ω}$ and satisfies $∥ \tilde{v} ∥_{L^{1} (R^{n})} + T V (\tilde{v}; R^{n}) \leq C_{Ω} (∥ v ∥_{L^{1} (Ω)} + T V (v; Ω))$ ; translations satisfy $∥ w (\cdot + h) - w ∥_{L^{1} (R^{n})} \leq ∣ h ∣ T V (w; R^{n})$ for $w \in B V (R^{n})$ ; and, for every open $U \subseteq R^{n}$ and every $u \in B V (U)$ , if $E_{t}^{+} : = {x \in U : u (x) > t}$ for $t > 0$ and $E_{t}^{-} : = {x \in U : u (x) < t}$ for $t < 0$ , then these sets have finite perimeter for almost every applicable $t$ and $T V (u; U) = \int_{0}^{\infty} P (E_{t}^{+}; U) d t + \int_{- \infty}^{0} P (E_{t}^{-}; U) d t .$

Statement references

The BV zero-extension and translation conclusions.

The signed-level BV coarea identity.

Facts & Assumptions

Given: Countable Choice, bounded Lipschitz domains, and BV functions on the stated domains.

[F1]

Bounded zero extension and the full-space translation estimate hold under Countable Choice, including dimension one. (BV extension and translation bounds)

[F2]

The signed-level coarea identity and almost-everywhere finite perimeter hold under Countable Choice on arbitrary open sets. (Coarea formula for spatial BV)

Proof

Given: The hypotheses in the Statement.

1.1F1given

Apply [F1] to each bounded Lipschitz domain and to the stated full-space BV functions. This proves the support, norm and translation assertions.

2.1F2givenstep 1.1∎

Apply [F2] on the arbitrary open set $U$ to $u \in B V (U)$ . It gives exactly the two signed level families, their almost-everywhere finite perimeters, and the displayed equality. All suppliers assume only Countable Choice.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 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

Gláucio Terra, Introduction to Geometric Measure Theory — Lecture Notes, version 1.6 (standard reference, not scraped)
Piotr Rybka, The space in variational and evolution problems — Lecture Notes in Mathematical Sciences 15 (standard reference, not scraped)