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.

One triangle computed by both the determinant and base--height formulas

Example

For $A = (1, 1)$ , $B = (5, 1)$ , and $C = (3, 4)$ , the triangle has Jordan content $6$ by both the determinant and base--height formulas.

Facts & Assumptions

Given: The three displayed vertices.

[L1]

Every triangle has content $\frac{1}{2} ∣ \det [B - A C - A] ∣$ , and for $A \neq B$ this equals half its base times perpendicular height (A triangle has content $\frac{1}{2} ∣ \det [B - A C - A] ∣$ , equal to half base times height when the chosen side is nonzero).

Verification

technique · direct

1.1

Here $B - A = (4, 0)$ and $C - A = (2, 3)$ , so $\det [B - A C - A] = 12$ and [L1] gives content $12 / 2 = 6$ .

L1algebra

2.1

The chosen horizontal base has length $4$ , and $C$ is at perpendicular distance $3$ from its line, so the base--height form in [L1] gives $\frac{1}{2} \cdot 4 \cdot 3 = 6$ again.

L1algebra∎

Depends on

A triangle has content $\tfrac12|\det[B-A\ C-A]|$, equal to half base times height when the chosen side is nonzero

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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.