Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04
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 symmetric monoidal category in which every object is self-dual

Example

The symmetric monoidal category of finite-dimensional vector spaces over a field has the property that every object is isomorphic to its dual.

Facts & Assumptions

Given: A finite-dimensional vector space V.

[L1]

The dual object V∗ exists in the category (Finite-dimensional vector spaces are rigid).

[L2]

dim⁡(V∗)=dim⁡(V) (The dual family of a finite basis is a basis of the dual space, with the same dimension).

[L3]

Equal finite dimensions imply isomorphism (Two finite-dimensional vector spaces over F are linearly isomorphic if and only if they have the same dimension).

Verification

technique · direct
1.1

By Finite-dimensional vector spaces are rigid, every finite-dimensional vector space V has dual object V∗.

givenL1
2.1

By The dual family of a finite basis is a basis of the dual space, with the same dimension, dim⁡(V∗)=dim⁡(V). Therefore Two finite-dimensional vector spaces over F are linearly isomorphic if and only if they have the same dimension gives an isomorphism V≅V∗.

step 1.1L2L3
3.1

Thus every object is self-dual up to isomorphism. The example is still only objectwise: the choice of isomorphism is generally noncanonical.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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