ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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 defines the writer monad by adjoining an accumulated output

Example

For a monoid $(M, e, \cdot)$ , define $T (X) = X \times M$ . The resulting writer monad carries a value together with an accumulated element of $M$ .

Facts & Assumptions

Given: A monoid $(M, e, \cdot)$ .

[L1]

The element $e$ is a two-sided identity and multiplication in $M$ is associative (Semigroup and monoid).

[L2]

A monad consists of a functor, a natural unit, and a natural associative multiplication satisfying the unit laws (Monad on a category).

Verification

technique · direct

1.1L1L2

Define $T (f) = f \times 1_{M}$ , $η_{X} (x) = (x, e)$ , and $μ_{X} ((x, m), n) = (x, m n)$ . These formulas have the required types.

2.1L1L2step 1.1

Naturality follows because the formulas do not change the $M$ -coordinate except by multiplication. The two monad unit composites send $(x, m)$ to $(x, e m)$ and $(x, m e)$ , both equal to $(x, m)$ by [L1].

3.1L1L2step 1.1∎

On an element $(((x, m), n), p)$ , the two associativity composites produce $(x, (m n) p)$ and $(x, m (n p))$ , equal by associativity in $M$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

E. Riehl, Category Theory in Context, 2nd ed., Example 5.1.5(iii) (standard reference, not scraped)