ExampleConstruction: AI-generatedVerification: AI-generatedSession-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.

Three collinear vertices give a triangle of zero content

Example

For $A = (0, 0)$ , $B = (2, 1)$ , and $C = (6, 3)$ , the triangle is the segment $[A, C]$ and has Jordan content $0$ .

Facts & Assumptions

Given: The three displayed vertices.

[L1]

A triangle has zero Jordan content if and only if its vertices are collinear (A triangle has zero Jordan content if and only if its vertices are collinear).

Verification

technique · direct

1.1

One has $C - A = 3 (B - A)$ , so the three vertices are collinear.

algebra

2.1

By [L1] the content is zero; directly, the determinant of $(2, 1)$ and $(6, 3)$ is $6 - 6 = 0$ , and the perpendicular height is zero as well.

step 1.1L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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.