ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

Kernels, cokernels, images, and coimages in abelian groups are the familiar subgroup and quotient constructions

Example

For a homomorphism $f : G \to H$ of abelian groups, the kernel is the usual subgroup $\ker (f) \subseteq G$ , the cokernel is the quotient $H / im (f)$ , the coimage is $G / \ker (f)$ , and the image is the subgroup $im (f) \subseteq H$ . The canonical map $G / \ker (f) \to im (f)$ is the usual first-isomorphism map $g + \ker (f) \mapsto f (g)$ .

Facts & Assumptions

Given: A homomorphism $f : G \to H$ of abelian groups.

[L1]

Abelian groups form an abelian category (Abelian groups form an abelian category).

Verification

technique · direct

1.1L1

In $A b$ , kernels and cokernels are computed by the familiar subgroup and quotient constructions. So the categorical kernel and cokernel of $f$ are exactly $\ker (f)$ and $H / im (f)$ .

2.1L1step 1.1∎

Therefore the categorical coimage is $G / \ker (f)$ and the categorical image is $im (f)$ . The canonical coimage-to-image comparison is the map $g + \ker (f) \mapsto f (g)$ , which is the usual first-isomorphism map.

Depends on

Abelian groups form an abelian category

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Gautam Tamme, Algebra II Lecture 9, §9.4 (standard reference, not scraped)