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CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Labelled unbracketed and unordered tensor strings are well defined in a symmetric monoidal category

Statement

In a symmetric monoidal category, once a finite tensor product is regarded as a labelled list of occurrences X1,…,Xn, it may be written without specifying either a bracketing or an order: any two parenthesised reorderings of that same labelled list are canonically and uniquely identified.

Facts & Assumptions

Given: A symmetric monoidal category and a labelled finite list of tensor factor occurrences.

[L1]

Symmetric coherence supplies a unique canonical natural isomorphism between any two parenthesised reorderings of the same finite list of labelled factor occurrences (Symmetric coherence).

Proof

technique · direct
1.1

Any two written forms of the same labelled tensor expression differ only by a choice of brackets and a permutation of the listed occurrences.

givenalgebra
2.1

By [L1], those two parenthesised reorderings are canonically and uniquely isomorphic. Therefore suppressing both the brackets and the order does not change the resulting labelled tensor expression except by that unique canonical identification.

L1step 1.1∎

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