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.

A bilinear form as a type $(0, 2)$ tensor

Example

On $R^{2}$ , the dot product

g ((a, b), (c, d)) : = a c + b d

is a type $(0, 2)$ tensor.

Facts & Assumptions

Given: The bilinear form $g$ on $R^{2}$ defined above.

[F1]

A type $(0, 2)$ tensor is a bilinear map $V \times V \to R$ (A type $(r, s)$ tensor on a finite-dimensional vector space).

Verification

technique · direct

1.1

The displayed formula is linear in $(a, b)$ and in $(c, d)$ separately, so $g$ is bilinear.

givenalgebra

2.1

By [F1], bilinearity is exactly the requirement for a type $(0, 2)$ tensor. Hence $g$ is such a tensor.

F1step 1.1

3.1

Therefore the Euclidean dot product is a concrete type $(0, 2)$ tensor.

step 2.1∎

Depends on

A type $(r,s)$ tensor on a finite-dimensional vector space

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)

A bilinear form as a type (0,2)(0,2) tensor