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 monoid object written with and without associators

Example

Let $(M, μ, η)$ be a monoid object in a monoidal category. Its associativity axiom can be displayed either with explicit bracket-changing data or with the unbracketed shorthand licensed by coherence.

Facts & Assumptions

Given: A monoid object $(M, μ, η)$ .

[L1]

The definition of monoid object writes the associativity and unit diagrams with explicit associator and unitor morphisms (Monoid objects and comonoid objects in a monoidal category).

[L2]

After coherence, those axioms may be written without explicit associators (The monoid-object axioms may be written without associators).

Verification

technique · direct

1.1

The explicit associativity axiom is $μ \circ (μ \otimes 1_{M}) = μ \circ (1_{M} \otimes μ) \circ α_{M, M, M}$ , and the unit axioms are $μ \circ (η \otimes 1_{M}) = λ_{M}$ and $μ \circ (1_{M} \otimes η) = ρ_{M}$ .

1.2

By [L2], the same relations may be written as $μ (μ \otimes 1_{M}) = μ (1_{M} \otimes μ)$ and $μ (η \otimes 1_{M}) = 1_{M} = μ (1_{M} \otimes η)$ , with the bracket suppressions understood canonically.

2.1

Thus the two notations express the same monoid-object structure; the second is shorter only because coherence has already absorbed the bracket changes.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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, Categories for the Working Mathematician, Chapter VII.3 (standard reference, not scraped)