ExampleConstruction: Literature-sourcedVerification: 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.

The two routes around the pentagon are equal

Example

For objects $A, B, C, D$ in a monoidal category, the pentagon compares two canonical composites from $((A \otimes B) \otimes C) \otimes D$ to $A \otimes (B \otimes (C \otimes D))$ .

Facts & Assumptions

Given: Four objects $A, B, C, D$ in a monoidal category.

[L1]

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

Verification

technique · direct

1.1

One route around the pentagon is the composite $((A \otimes B) \otimes C) \otimes D \overset{α_{A \otimes B, C, D}}{\to} (A \otimes B) \otimes (C \otimes D) \overset{α_{A, B, C \otimes D}}{\to} A \otimes (B \otimes (C \otimes D))$ .

given

1.2

The other route is the composite $((A \otimes B) \otimes C) \otimes D \overset{α_{A, B, C} \otimes 1_{D}}{\to} (A \otimes (B \otimes C)) \otimes D \overset{α_{A, B \otimes C, D}}{\to} A \otimes ((B \otimes C) \otimes D) \overset{1_{A} \otimes α_{B, C, D}}{\to} A \otimes (B \otimes (C \otimes D))$ .

given

2.1

Both composites are canonical morphisms with the same source and target, so [L1] makes them equal.

L1step 1.1step 1.2∎

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

S. Mac Lane, Natural Associativity and Commutativity, equation (3.5) (standard reference, not scraped)