CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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 volume under a nonnegative continuous graph over a compact Jordan base is its integral

Statement

Let $D \subseteq R^{m}$ be compact and Jordan measurable and let $g : D \to [0, \infty)$ be continuous. Then $U_{g} : = {(u, t) : u \in D, 0 \leq t \leq g (u)}$ is compact and Jordan measurable, and

$cont (U_{g}) = \int_{D} g (u) d u .$

Facts & Assumptions

Given: The set $D$ , the nonnegative continuous function $g$ , and the constant integrand $H = 1$ on $U_{g}$ .

[F1]

If $D \subseteq R^{m}$ is compact and Jordan measurable and $α, β : D \to R$ are continuous with $α \leq β$ , then $K = {(u, t) : u \in D, α (u) \leq t \leq β (u)}$ is compact and Jordan measurable and every continuous $H : K \to R$ satisfies $\int_{K} H = \int_{D} (\int_{α (u)}^{β (u)} H (u, t) d t) d u$ (A solid between continuous graphs over a compact Jordan base is Jordan measurable and integrates by vertical sections).

Proof

technique · direct

1.1

Apply [F1] with lower graph $0$ , upper graph $g$ , and integrand $H = 1$ . This includes the empty base and the identically zero graph.

givenF1

2.1

With $H = 1$ the left side of the formula in [F1] is $\int_{U_{g}} 1 = cont (U_{g})$ and the inner integral is $\int_{0}^{g (u)} 1 d t = g (u)$ , so that formula becomes $cont (U_{g}) = \int_{D} g$ .

step 1.1F1algebra∎

Depends on

A solid between continuous graphs over a compact Jordan base is Jordan measurable and integrates by vertical sections

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Sigurd Angenent, Math 221 lecture notes, Chapter 8 §3.2 (standard reference, not scraped)