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.

Tensor product and contraction in a basis

Example

On $V = R^{2}$ with basis $(e_{1}, e_{2})$ and dual basis $(e^{1}, e^{2})$ , let

S = e^{1}, T = e^{2}, R = e_{1} \otimes e^{1} + e_{2} \otimes e^{2} .

Then

S \otimes T = e^{1} \otimes e^{2}, ctr (R) = 2.

Facts & Assumptions

Given: The basis $(e_{1}, e_{2})$ , its dual basis $(e^{1}, e^{2})$ , and the tensors $S, T, R$ above.

[L1]

Tensor product multiplies the factor values on concatenated arguments, and contraction is basis-independent (Tensor product of multilinear tensors is associative and bilinear, Contraction is independent of the basis formula).

Verification

technique · direct

1.1

For vectors $(a_{1}, a_{2})$ and $(b_{1}, b_{2})$ , [L1] gives $(S \otimes T) ((a_{1}, a_{2}), (b_{1}, b_{2})) = S (a_{1}, a_{2}) T (b_{1}, b_{2}) = a_{1} b_{2},$ so $S \otimes T = e^{1} \otimes e^{2}$ .

L1givenalgebra

1.2

Again by [L1], $ctr (R) = R (e^{1}, e_{1}) + R (e^{2}, e_{2}) = 1 + 1 = 2.$

L1givenalgebra

2.1

This computes the announced tensor product and contraction.

step 1.1step 1.2∎

Depends on

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.

Sources

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