ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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 canonical free-algebra presentation of a two-element idempotent monoid

Example

Let $M = {1, e}$ be the monoid with $1$ as identity and $e^{2} = e$ . For the free-monoid monad, its canonical presentation is

$(M^{*})^{*} \begin{array}{c} \overset{T (a)}{\to} \\ \to_{μ_{M}}^{} \end{array} M^{*} \overset{a}{\to} M,$

where $a$ evaluates a word in $M$ , $T (a)$ evaluates each inner word, and $μ_{M}$ concatenates the inner words.

Facts & Assumptions

Given: The two-element monoid $M$ and the three displayed word maps.

[L1]

Every $T$ -algebra is the coequalizer of its canonical pair of free algebras (Every algebra is the coequalizer of its canonical pair of free algebras).

[L2]

The free-monoid monad inserts letters as one-letter words and flattens words of words by concatenation (The free-monoid monad has monoids as its Eilenberg–Moore algebras).

Verification

technique · direct

1.1construct

The multiplication table is $1 \cdot 1 = 1$ , $1 \cdot e = e \cdot 1 = e$ , and $e \cdot e = e$ , so $M$ is a monoid and $a : M^{*} \to M$ evaluates every finite word to its product.

1.2L1L2construct

On a word of words $[w_{1}, \dots, w_{n}]$ , the map $T (a)$ gives $[a (w_{1}), \dots, a (w_{n})]$ , while $μ_{M}$ gives the concatenated word $w_{1} \dots w_{n}$ , as in [L1] and [L2].

2.1step 1.1step 1.2algebra

Evaluating either result multiplies the same letters in the same order, so $a T (a) = a μ_{M}$ for every finite word of words, including the empty one and words containing empty inner words.

3.1step 2.1L1∎

The theorem [L1] now gives the coequalizer universal property. On underlying sets the sections are the one-letter-word maps $η_{M^{*}}$ and $η_{M}$ , and the monad unit and naturality equations verify the split presentation.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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.