ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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 disc has Jordan content $π$

Example

The unit disc $D_{1} = {(x, y) : x^{2} + y^{2} \leq 1}$ is Jordan measurable and has content $π$ .

Facts & Assumptions

Given: The closed disc of radius $1$ .

[L1]

A closed disc of radius $r \geq 0$ has Jordan content $π r^{2}$ (A closed disc of radius $r \geq 0$ has Jordan content $π r^{2}$ ).

[L2]

A region between continuous graphs is compact and Jordan measurable, and its graph area equals its Jordan content (Riemann area between continuous graphs equals Jordan content).

Verification

technique · direct

1.1

Substitute $r = 1$ in [L1] to obtain $cont (D_{1}) = π$ .

L1algebra

2.1

The same disc lies between $y = - \sqrt{1 - x^{2}}$ and $y = \sqrt{1 - x^{2}}$ on $[- 1, 1]$ , so [L2] identifies that value with the established graph-area convention, including the endpoint sections of height zero.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

W. F. Trench, Introduction to Real Analysis, §§7.2–7.3 (standard reference, not scraped)

The unit disc has Jordan content ππ