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.
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.

Wedge products of the standard dual basis

Example

On $R^{3}$ with standard dual basis $d x, d y, d z$ ,

$d x \land d y = - d y \land d x, d x \land d x = 0, d x \land d y \land d z$

is a basis of ${Alt}^{3} (R^{3})$ .

Facts & Assumptions

Given: The standard basis of $R^{3}$ and its dual basis $d x, d y, d z$ .

[L1]

The wedge product is graded commutative (The wedge product is associative and graded commutative).

[L2]

Increasing wedge monomials in a dual basis form a basis of each exterior-power space (Wedge monomials in a dual basis form a basis).

Verification

technique · direct

1.1

Since $d x$ and $d y$ are $1$ -forms, [L1] gives $d x \land d y = - d y \land d x$ and $d x \land d x = - d x \land d x$ , hence $d x \land d x = 0$ .

L1givenalgebra

1.2

By [L2], the unique increasing triple wedge $d x \land d y \land d z$ forms a basis of ${Alt}^{3} (R^{3})$ .

L2given

2.1

These are the standard wedge-product identities in the dual basis.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)