ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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 canonical map between two bracketings of a five-fold product

Example

Take the two parenthesisations $u : = ((A \otimes (B \otimes C)) \otimes (D \otimes E)), v : = A \otimes (B \otimes (C \otimes (D \otimes E))) .$ A canonical map $u \to v$ is obtained by associators alone.

Facts & Assumptions

Given: Objects $A, B, C, D, E$ of a monoidal category.

[L1]

Between any two parenthesisations of one ordered tensor word there is a unique canonical natural isomorphism (Mac Lane coherence in canonical-map form).

Verification

technique · direct

1.1given

Apply the outer associator to obtain $((A \otimes (B \otimes C)) \otimes (D \otimes E)) \overset{α_{A, B \otimes C, D \otimes E}}{\to} A \otimes ((B \otimes C) \otimes (D \otimes E))$ .

2.1step 1.1

Apply $1_{A} \otimes α_{B, C, D \otimes E}$ to the result of step 1.1 and get $A \otimes ((B \otimes C) \otimes (D \otimes E)) \to A \otimes (B \otimes (C \otimes (D \otimes E))) = v$ .

3.1L1step 2.1∎

The composite of steps 1.1 and 2.1 is therefore a canonical map $u \to v$ . Any other chain of associators with the same endpoints is equal to it by [L1].

Depends on

Mac Lane coherence in canonical-map form

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Chapter 2.9 (standard reference, not scraped)