ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01

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.
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The identity endomorphism and its coordinate-independent trace

Example

For the identity endomorphism ${id}_{R^{n}}$ , the associated type $(1, 1)$ tensor has contraction $n$ , so its trace is $n$ in every basis.

Facts & Assumptions

Given: A basis $e_{1}, \dots, e_{n}$ of $R^{n}$ with dual basis $e^{1}, \dots, e^{n}$ , and the identity endomorphism.

[F1]

Contraction is the dual-basis sum $\sum_{i} T (e^{i}, e_{i})$ (The contraction of a mixed tensor).

[F2]

Trace of an endomorphism is the sum of the diagonal entries in any basis (The basis-independent trace of an endomorphism of a finite-dimensional vector space).

Verification

technique · direct

1.1

The tensor associated to ${id}_{R^{n}}$ is $T (α, v) = α (v)$ . Therefore [F1] gives $ctr (T) = \sum_{i = 1}^{n} e^{i} (e_{i}) = n .$

F1givenalgebra

2.1

In the chosen basis, the matrix of the identity endomorphism is the identity matrix, so [F2] gives trace $n$ . This agrees with step 1.1, and neither value depends on the basis.

F2step 1.1algebra

3.1

Thus the identity endomorphism has coordinate-independent trace $n$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Will J. Merry, Differential Geometry (standard reference, not scraped)