ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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 content of a parallelogram from a two-by-two matrix

Example

The parallelogram spanned by $v = (3, 1)$ and $w = (1, 3)$ has Jordan content $8$ .

Facts & Assumptions

Given: The displayed spanning vectors and base point $0$ .

[L1]

A parallelogram has content $∣ \det [v w] ∣$ , equal to base times height when $v \neq 0$ (A parallelogram has Jordan content $∣ \det [v w] ∣$ , equal to base times height when $v \neq 0$ ).

Verification

technique · direct

1.1L1algebra

The determinant is $3 \cdot 3 - 1 \cdot 1 = 8$ , so [L1] gives content $8$ .

2.1L1algebra∎

The base length is $\sqrt{10}$ . Since $⟨ v, w ⟩ = 6$ , the residual after projecting $w$ onto $R v$ is $(- 4 / 5, 12 / 5)$ and has norm $8 / \sqrt{10}$ ; the base--height product is therefore $\sqrt{10} \cdot 8 / \sqrt{10} = 8$ .

Depends on

A parallelogram has Jordan content $|\det[v\ w]|$, equal to base times height when $v\ne0$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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.