ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Building an upper-triangular matrix from a complete invariant flag

Example

On $Q^{3}$ , define $T e_{1} = 2 e_{1}, T e_{2} = e_{1} + 3 e_{2}, T e_{3} = 2 e_{1} - e_{2} + 4 e_{3} .$ The complete flag $0 \subset ⟨ e_{1} ⟩ \subset ⟨ e_{1}, e_{2} ⟩ \subset Q^{3}$ is $T$ -invariant, and in the flag-adapted basis $(e_{1}, e_{2}, e_{3})$ , $[T] = (\begin{array}{ccc} 2 & 1 & 2 \\ 0 & 3 & - 1 \\ 0 & 0 & 4 \end{array}) .$

Facts & Assumptions

Given: The displayed action of $T$ on the standard basis of $Q^{3}$ .

[L1]

A basis gives an upper-triangular matrix exactly when its successive spans form a complete invariant flag (Complete invariant flags are equivalent to upper-triangular matrices).

Verification

technique · computation

1.1algebra

The formulas show $T ⟨ e_{1} ⟩ \subseteq ⟨ e_{1} ⟩$ and $T ⟨ e_{1}, e_{2} ⟩ \subseteq ⟨ e_{1}, e_{2} ⟩$ , while the final space is automatically invariant.

2.1step 1.1L1algebra∎

The coordinate columns of $T e_{1}, T e_{2}, T e_{3}$ give the displayed matrix, and [L1] identifies this upper-triangular form with the verified complete flag.

Depends on

Complete invariant flags are equivalent to upper-triangular matrices

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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.