ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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.

The list monad on a two-element set

Example

For $X = {a, b}$ , the list monad gives

$T (X) = {[], [a], [b], [a, a], [a, b], [b, a], [b, b], \dots} .$

The unit inserts a letter as a one-letter word, and multiplication removes one level of list brackets by concatenation.

Facts & Assumptions

Given: The set $X = {a, b}$ .

[L1]

The free-monoid monad sends a set to its finite words, has singleton-word unit, and has concatenation multiplication (The free-monoid monad has monoids as its Eilenberg–Moore algebras).

Verification

technique · direct

1.1L1

The displayed beginning contains the empty word, the words $[a], [b]$ , and the words $[a, a], [a, b], [b, a], [b, b]$ ; $T (X)$ continues with all finite words in $a$ and $b$ .

2.1L1step 1.1

One has $η_{X} (a) = [a]$ , $η_{X} (b) = [b]$ , $μ_{X} ([[a, b], [], [b],]) = [a, b, b]$ , $μ_{X} ([]) = []$ , and $μ_{X} ([[]]) = []$ .

3.1L1step 2.1∎

The unit samples flatten $[[a],]$ to $[a]$ and the list of singleton words to the original word; nested concatenation gives the same word under either bracketing. Thus the displayed calculations agree with the general monad equations.

Depends on

The free-monoid monad has monoids as its Eilenberg–Moore algebras

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.