ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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 unit box in $\mathbb{R}^m$ has volume $1$ , and the integral of a constant $c$ over it is $c$

Example

For $Q=[0,1]^m$ , $\operatorname{vol}(Q)=1$ , and for every constant $c$ , $\int_Qc=c$ .

Facts & Assumptions

Given: $m\ge1$ and constant $c$ .

[L1]

Rectangle volume is the finite product of side lengths (Axis-parallel rectangles in $\mathbb{R}^m$ and their volume).

[L2]

Darboux sums and the integral are Lower and upper Darboux sums over a grid partition in $\mathbb{R}^m$ and The lower and upper Darboux integrals over a nondegenerate rectangle in $\mathbb{R}^m$ .

Verification

technique · direct

1.1

Every side has length $1$ , so the finite product volume is $1$ .

L1given

1.2

On every cell, both infimum and supremum of the constant function are $c$ . Thus both sums are $c\sum_i\operatorname{vol}(Q_i)=c$ .

L2given

2.1

Lower and upper integrals therefore both equal $c$ .

step 1.2L2∎

Depends on

Used by

The rational points of [0,1]² form a bounded null set that is not Jordan measurable Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 70 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Lebl, Basic Analysis, Jordan Measurable Sets (standard reference, not scraped)
J. Lebl, Basic Analysis, Outer Measure and Null Sets (standard reference, not scraped)
J. Lebl, Basic Analysis, Riemann Integral in Several Variables (standard reference, not scraped)

The unit box in Rm\mathbb{R}^mRm has volume 111, and the integral of a constant ccc over it is ccc