ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-29

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.

An ordinary element as the member from the integers

Example

Let $A$ be an abelian group and let $a \in A$ . The homomorphism $x_{a} : Z \to A, x_{a} (n) = n a,$ is a member of $A$ in the sense of Member of an object. Its associated subobject is the cyclic subgroup generated by $a$ .

Facts & Assumptions

Given: An abelian group $A$ and an element $a \in A$ .

[L1]

A member of an object is just a morphism into it (Member of an object).

[L2]

Member classes correspond to subobjects (Members modulo equivalence correspond to subobjects).

[L3]

The category $A b$ is abelian (Abelian groups form an abelian category).

Verification

technique · direct

1.1L1L3

The map $x_{a}$ is a group homomorphism $Z \to A$ , hence a member of $A$ by [L1].

2.1L2step 1.1∎

Its image is the subgroup $⟨ a ⟩$ , so [L2] identifies the corresponding subobject with the cyclic subgroup generated by $a$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Saunders Mac Lane, Categories for the Working Mathematician, VIII.4 (standard reference, not scraped)